There is no quick way to improve your chess. However, the following would definitely help.
Having said all that, tho' here a few more specific hints.
First, a quick primer on the relative values of the pieces. These values only have meaning when deciding whether or not to trade pieces. One is not necessarily winning just because one has more material. Having said that, here it is:
Pawn = 1 Knight = 3 Bishop = 3.25 Rook = 5 Queen = 9
The King is never actually captured, and thus is not listed. The Bishop, while slightly more valuable than a Knight in most cases, is often traded on an equal basis with the Knight. Two Bishops, however, is almost always better than Two Knights, as the advantage of the Bishops is additive. In fact, one place they are definitely superior is in the endgame. You can force checkmate with Two Bishops vs. a lone King, but cannot _force_ checkmate with two Knights vs. a lone king.
Also, in chess parlance, the Pawn is not considered a "piece" - Pawns and pieces are usually referred to as separate things. Knights and Bishops are considered minor pieces, Rooks and Queens major pieces.
Having defined a few terms, on to the guidelines.
1. Start with moving a center pawn 2 squares forward. This occupies and controls the center, meanwhile opening lines for your pieces to get into play (in chess parlance, we say the pieces are "developed" when brought into play). This also allows you to keep the side (or "wing") pawns intact so your King has a safe place to castle later. This rule is here because center-pawn openings are both the easiest to play and to understand, and should be concentrated on by the beginning player. It does not mean other choices are bad, they're just not good ones for beginning players.
2. Make only as many Pawn moves as is necessary in the opening to effectively develop the rest of your pieces or as necessary for defense. This is usually no more than four in the early opening phase of the game. The end of the opening phase is generally considered to be reached when all the pieces have been developed and the King has castled.
3. Get all your pieces developed as quickly as possible. One piece will not accomplish anything by itself, and you basically can't do anything until you do this first.
4. Develop Knights before Bishops. Knights move more slowly than Bishops, and take longer to get where they're going. A Bishop can travel long range and be developed almost instantly to the desired square.
5. Develop Knights toward the center (not the edges). Remember, "A Knight on the rim is dim." This is because it only has half the scope there as in the center. You can prove this by putting a Knight in the center of the board, and counting the squares it attacks (you should get eight), and then putting it on the side, and counting how many squares it covers (four). Since this placement has a larger effect percentage-wise on the Knight than any other piece (try it - and notice what happens to the Rook!), and the Knight is also a slow moving piece, this rule actually applies to almost all pieces, but especially the Knight.
6. Castle early for King safety and to develop the Rooks, usually immediately after the development of Knights and Bishops, sometimes even before both Knights and both Bishops have come into play, depending on necessity. If you can wait until the Knights and Bishops are in play, however, this gives you the choice of deciding on which side to castle, though sometimes there is only one good choice, as one wing or the other may already be weakened, in which case delaying would give no benefit. See items #1 and #21.
7. Do not move a piece twice in the opening. Doing so delays getting your other pieces developed and delays castling.
8. Do not bring the Queen out too early. It is a valuable piece, and therefore an easy target for lesser pieces. If you bring it out too early, you are likely to find yourself moving the Queen over and over to get it to safety (repeatedly violating guideline #7), and delaying the development of your pieces while your opponent essentially takes one free turn after another, developing his. If you need to move it so your King can castle queenside, generally moving it up to the second rank just to get it out of the way is ok. Moving it farther than this before you've finished development is usually inviting trouble.
9. Control the center squares. Traffic generally has to run through the center of the board in one way or another. Control the center of the board, and you usually will have more freedom to put your plans into effect than your opponent, as your pieces will have more scope &power from the center of the board, being able to get to any spot on the board relatively quickly.
10. Keep one or more Pawns in the center. This helps you achieve #9 above, as one way to control the center is by occupying it, and Pawns are the most difficult piece to budge.
11. Place your pieces on open lines (open lines are lines of movement that are unobstructed by pawns). Place Bishops on open diagonals, Rooks on open files (files are columns, ranks are rows).
12. Coordinate your pieces to work together. One common way to do this is to double pieces up, such as putting a Queen and Bishop on the same diagonal, or putting two Rooks on the same file or rank. They support each other's movement along the diagonal, file, or rank in question, and are essentially twice as powerful this way.
13. When protecting a piece, use the least valuable piece available to do so. Especially, protect Pawns with Pawns (forming a Pawn chain). Why tie up a valuable piece to protect a Pawn if it's not necessary? Not only does using less valuable pieces for protection free up the more powerful pieces, but the less valuable pieces are less likely to be scared away or dislodged from their defensive posts.
14. Avoid isolated Pawns if possible. Isolated pawns are those that can no longer be protected by an adjacent pawn (because there are no pawns on the adjacent files). Isolated pawns generally occur as a result of Pawn captures being made, so carefully look at the resulting Pawn structure when you have the choice of capturing with a Pawn or another piece. Isolated pawns are weak because they are subject to attack and must be defended by other pieces.
15. Especially avoid doubled isolated Pawns. Doubled Pawns are two pawns of the same color on the same file. The Pawn in back is weak because its movement is inhibited by the Pawn in front. This is not terribly weak by itself. However, doubled isolated Pawns are _very_ weak, as they both cannot be supported by other pawns and cannot move freely, a bad combination.
16. Make moves that threaten, when possible. These moves limit your opponent's choices, and basically allow you to call the shots, as your opponent usually must respond to your threat before proceeding with his own threats. Alternately, do not get carried away with making your own threats to the point that you overlook your opponent's threats. Being the one who is calling the shots is called having the initiative.
17. Don't make pointless threats. This includes checks. Checking or attacking something simply for the sake of doing so has no value. If the threatened piece can simply move away with no detrimental consequences, and there is no advantage to you in making the move in the first place, then the threat is pointless. Doing this can even force your opponent to make a good move. Pointlessly threatening a Knight on the rim just forces your opponent to move it back towards the center of the board, for example. However, if the Knight is trapped there, then attacking it would allow you to win the piece.
18. When ahead in material, exchange pieces. For example, if the total value of your pieces on the board (see relative values listed above) is 16, and the total value of your opponents pieces is 11, this is roughly a 3 to 2 edge. Trade Rooks, however, and now the total value of your pieces is 11, your opponent's pieces have a total value of 6. This is almost a 2 to 1 edge, which is obviously better.
19. When behind in material, don't exchange pieces. This is essentially #18 looked at from the other side of the coin.
20. When you are attacked, try to exchange the attacking pieces to reduce the power of the attack. This takes precedence over #18 & #19, as the safety of the King is more important than anything else.
21. Don't weaken Pawns in front of your castled King. Generally this means don't move them unless you absolutely have to. Once moved forward they become easier targets for attack.
22. Try not to leave your pieces in positions where they are loose (undefended). Loose pieces become targets for attack, and are more likely to be lost than pieces that are defended.
23. Avoid creating holes in your position. A hole is a square that can no longer be defended by a Pawn. Since a Pawn is the most useful piece when it comes to threatening another piece to make it move away from a particular spot, this would mean a piece can lodge itself in this hole and be extremely hard to drive away. This is especially bad if the hole is near where your king is hiding out.
24. Bring your King into action in the endgame. Once the danger of the middle game is over and there are very few pieces on the board, the King need not cower in the corner anymore. He instead becomes a powerful attacking piece.
25. Find your opponent's weaknesses and exploit them. This may be anything from a set of doubled or isolated pawns to a vulnerable King position to something as esoteric (and beyond the scope of this file) as a weak square or a lack of development. To understand how to exploit these weaknesses, play over master games, and watch how they do it. Games of the old masters (Morphy, Tarrasch, Nimzovitch, Lasker, & Capablanca, for example) often illustrate these concepts better than modern games, as they are simply easier to follow and more straightforward in their style of play.
26. Don't sacrifice a piece without a clear reason, like a DEFINITE checkmate. Only masters are justified in making speculative sacrifices, and even they will not generally do so. Speculative sacrifices fail much more often than they succeed. If your name is Mikhail Tal, you can ignore this one.
27. Always assume your opponent will make the best move. Assume he will be fooled, and you will eventually set yourself up for trouble. Only by determining the best moves for _both_ sides can accurate analysis be done.
28. Do not follow any of these preceding guidelines blindly or mechanically. Analysis always supersedes these guidelines. If your analysis says you have checkmate in three moves no matter how your opponent replies (you are said to "have mate in three"), then all the guidelines go out the window. Obviously, it doesn't matter if you have to put your Knight on the rim to deliver checkmate. If you see a _specific_ reason to break a guideline, such as mate or the win of material, particularly of a piece or more, and you believe your analysis is sound and the benefits outweigh the negatives of breaking that guideline, then by all means break it. These guidelines are simply to help you win, and are not hard and fast rules for every situation. This leads us to #29...
29. Be careful when grabbing material, as it can be used as a way of luring your pieces into positions where they will be useless to prevent an onslaught against your King, or even in preventing your pieces from ever developing and getting into the game. But as Bobby Fischer once said (paraphrased), "If you can't see a good reason not to take a piece, then take it." Simply put, don't get greedy over material at the expense of the safety of your King, but don't refuse outright gifts, either.
30. And fittingly last, if you are playing a game and are a Rook down or more, with no attack, passed pawn (a Pawn whose passage is unopposed by other pawns and thus is a serious threat to promote to a queen), or other significant compensation, against a knowledgeable player who is not likely to blunder badly enough for you to get back in the game, graciously resign and get on with the next game. There are exceptions to this, especially with timed play and/or when tournament prize money is on the line at the amateur level, but it is generally a sign of good sportsmanship to admit when you've been defeated and congratulate your opponent. This almost always occurs at the master level, regardless of circumstance - few would be caught dead playing drearily on until the inevitable mate is delivered. It should _always_ occur when the games are friendly. Also, if you are the victor, be gracious about winning, don't gloat, and compliment your opponent on the things he did right.