Zombie costumes can be as simple or elaborate as you like. The one rule to follow is ONLY wear something that you intend to completely destroy.
Dig out some old clothes you don't like or hit thrift stores for something cheap.
Now for the fun. No respectable zombie is going to have perfectly washed and pressed clothes. You need to do something to make them look worn, torn, dirty, and blood-stained. The main thing is to use your imagination.
Rips & Tears: You're a zombie; you stumble into, through, and over things. At some point, you probably fought for your life before you were violently relieved of it. You need to figure out what damage your clothes have taken. Maybe sleeves or pockets were ripped loose by your attackers. Maybe you've gotten snagged on fences or bushes while stumbling around.
Try to actually rip the fabric. It'll have a much more natural look than if you cut it. If necessary, use scissors to start a small cut and rip the rest of the tear that you want.
Dirt & Filth: You can paint your clothes with mucky looking colors or preferably you can actually get them dirty. Go roll around in the grass and dirt. Clover patches are good for a nice grass stain. Try spraying a little water on your clothes first so the dirt will stick better.
Blood: This one's completely dependent on your personal taste. A lot or a little. Paint it on ahead of time or dribble some on as a final touch once you're suited up and in makeup. It's up to you.