The first step in playing for a draw is opening preparation. Ideally you know what your opponent usually plays and can prepare something against that. The key, obviously, is to pick calm openings with little in the way of tactics. If you can get the queens off early so much the better.
Kramnik gave the ultimate example in his match with Kasparov when he used the Berlin Wall variation of the Ruy Lopez. Time and again he quickly reached a queenless middlegame with simple, clear themes and a clear plan to achieve and maintain equality.
In general aim to strengthen your position rather than launch attacks. Pay attention to Nimzowitschian themes like over protecting the center, finding good squares for your pieces, completing your development harmoniously, putting your king safe by castling. Of course you would normally aim to do these things anyway (wouldn't you?) but when aiming for a draw make these your priority before any tactical plans you might be tempted by.
When tactics do arise put the emphasis, where possible, on simplifying via exchanges rather than complicating the position. At the same time don't make the mistake of chasing exchanges. If your opponent is stronger he will not try and avoid exchanges completely but will instead try and make sure that if exchanges do occur then they are on his terms and his position is marginally improved by the exchange.
Be aware that, particularly in the endgame, playing for a draw does not mean playing passively. Unless you achieve a fortress position where your opponent simply can't make progress you will still have to take care in the endgame to play actively. Rooks are better attacking enemy weak pawns than defending yours if possible. Your king should be active aiming to attack enemy pawns if possible and opposing the enemy king when he tries to threaten your pawns rather than just passively defending. If you have a pawn majority on one side and a minority on the other try and push the majority and play on that side rather than move the other pawns except for absolutely necessary for defense.
Playing for a draw is a mindset and you mustn't let yourself be distracted by tactical ploys by your opponent.