Studying classic chess games is something that the former Soviet players and chess schools placed great emphasis on. It's certainly something you should be considering as part of your chess improvement programme.
Studying the games from the great players serves many purposes: you will learn more strategic plans; better understand how to recognise and exploit positional strengths and weaknesses; improve your pattern recognition and ability to form suitable plans; and have greater awareness of tactical opportunities.
My advice is to adopt the approach suggested by Dvoretsky (in Secrets of Chess Training): play through the opening, and then cover up one side's moves and try to guess them, spending at least 45 minutes on studying each game.
You will need the games available in a format that lets you cover up moves easily, and preferably you will want games that have been well annotated. We'll look at making a set of games available via Chess Tales in a suitable format.
Which games should you study? Dvoretsky and Kotov (in Think Like a Grandmaster) suggest beginning with Rubinstein (a master strategist and endgame player) and then moving on to Alekhine and Capablanca. Along with these 3, I suggest working back through the World Champions: Kasparov, Karpov, Fischer, etc.. An alternative is to get a well chosen list of games that has already been prepared, for example Nunn's Understanding Chess Move by Move.
Obviously time will determine how many games you can study, but even just one game per week would see you absorb 52 wonderful games of chess and new ideas by the end of a year.
Studying games like this will also have the side effect of helping you, as Kasparov recommends, to continually broaden and deepen your opening repertoire.