To build on that, there are a few things you want to bear in mind, all of which can mostly be boiled down to 'think about what you're doing'... the key really is not making rash moves, what seems like a free piece or a mistake by your opponent is just as likely to be a carefully planned trap if they know what they're doing. Think about all the variables, 4-5 moves ahead if you can, and take your time before committing to a move. With that in mind:
- As Nev said, protecting your pieces is vital. Ideally, you want a setup where every piece is guarded such that if the opponent moves to take it, the piece that does the taking will then get taken in return. Bear in mind though that your opponent will be trying to do the same so sometimes you'll need to set up multiple threats to the same square. If he has both a Bishop and a Knight threatening one of your pieces then one defence isn't going to be enough, as you'll move in to take the first piece and leave yourself open to being taken by the other model threatening the same square. Thus, you also need a second piece able to threaten that square, so he will be looking to find a third, so on and so forth.
Such rapid plays of take and counter-take can be made to work, if the gain you make suits your overall strategy, but should not be rushed into. Taking a handful of pieces in quick succession is no good if doing so leaves your defence open somewhere else on the board, or more likely just achieves nothing other than clearing up the board a bit. Every move you make should be made for a reason.
- Think before every move 'what will I gain from doing this?' It might be taking or threatening a piece, it might be forcing your opponent to move something they didn't want to, and thus opening a defence, it might be defending or blocking the way to one of your own pieces... Whatever the reason, think before you act. Furthermore, while it's often tempting to focus on the 'powerhouse' pieces, the Queen and Rooks (and arguably Bishops), they might not offer the most useful move. A Pawn in the right place is more useful than a Queen in the wrong one.
- Be prepared to play the 'long game', so long as you physically have the time. While a quick victory can be satisfying, and some players will find they are simply more comfortable or effective when playing aggressively, don't be afraid to use your opening turns to routinely set up a strong defensive position. The first 10 turns of any game I play tend to be using Knights and Pawns to lock down basically any available square on my side of the board, set up in such a way that my Queen and a Bishop can then strike out along the diagonals (this is something else to always consider in the early game, not blocking off the movement of your more dangerous pieces) and withdraw to a position of safety should the opportunity not be there. Don't worry about being predictable or repetitive in the early game, a common defence is still a solid one until your opponent figures out its weakness (at which point, you change it up and experiment with some different strategies).
- Moving on to offence, the main way you'll take pieces against an experienced opponent is not through them making mistakes, but through forcing them to make choices. Wherever possible, set up a situation where you're threatening multiple pieces with a single move, meaning that whatever your opponent does, they're likely to lose something. Even if you don't then follow through on such a threat (for instance, if it would expose one of your pieces), you have just dictated what he's used his turn to do. The ultimate form of this is threatening both the King and another valuable piece; the only move available to the opponent is to move the King out of check, leaving you free to take the other piece. This is definitely something you'll pick up simply through gaining experience.
- When it comes to the end game, when you've largely cleared the board, what you need to be thinking is 'what squares can the King move to, and how do I stop him getting there?' This is where thinking carefully about what squares you can threaten is key. A Bishop can only threaten squares the same colour as the one it starts on, so conversely you can use a Bishop on a black square to make any black square dangerous for the King, and the same for white. A Rook can shut down an entire row of squares, meaning you can force the King to move left or right by placing a Rook (or Queen) on the opposite side. If you can get two Rooks or a Rook and a Queen next to each other, you can 'leapfrog' them each turn to push the king across to the edge of the board, then close off the adjacent row, then threaten the final row with your other Queen/Rook, and that's Checkmate. There are far too many similar methods to get into here (that one is called the 'lawnmower', I believe) but the important thing is to remember that even in the late stages think about your moves, otherwise it's very easy to just get sucked into chasing the King around with no result until you run out of time.
Sorry, rambled on a bit there. As you can tell, I like chess!

The thing is, this might sound fiddly but the more games you play, the more it becomes second nature. Unlike most types of wargaming, there is no random element in Chess, so at any moment you control all the variables; if you can think more carefully and concentrate better than your opponent, and use your moves and pieces more effectively, you
will will. A move or method will only fail if you let it, by not taking the time to think it through, or because you missed something (which will happen, there are still a ton of variables even if they're not random). The key is practice, practice and when you think you're done practising, practice some more. Learn from every game, the ones you win, the ones you lose, and remember to have fun while you're doing it!