It depends upon what type of tires you use,whether they are fast wearing racing compound, or tougher touring tires, your weight, and the tarmac surfaces upon which you ride. A rough (very rough) rule would be: See to your racing tires after 1 000 miles of road riding, (or a weekend's racing) and your touring tires after anywhere from 2 500 to 4 000 miles. Some brands outlast others... Michelins will go on seemingly forever, although you may wish for their demise, as their grip is inferior to most other brands. Continentals, in my experience, grip better, but last half as long.
Examine the tires for signs of wear... Road tires that have ribs or tread patterns will wear smooth down the centre, Slick road tires will wear "flat" across the centre. Also, look for cracks, tears, small chunks missing on the tread surface. These are all signs of worn out bicycle tires.
If you detect wear on the rear tire, but the front looks okay, as is nearly always the case, you can dismount the tires and swap them, and run the better ex-front tire on the back.
Always save the best of your old tires. A worn spare is better than no spare.
Replace tires with ones of the same width, and, if you liked the originals, the same brand. You can experiment with slightly narrower or wider tires, but be advised... a few mm of width will noticeably change the bike's handling.
I just ordered another set of Performance GT2 Kevlar, in 700x25 size, for US$16ea. See them at
http://www.PerformanceBike.com. This makes perhaps five sets of these that I've used, and I average about 4 000 miles per set, rotating them once. The kevlar belt provides that little something extra against punctures. They are heavier than more expensive tires, and the herringbone tread means that they're not as fast as racing slicks, but they're exceptionally good at keeping air within and glass shards, nails, and road debris without, and that's what counts for me.
One can buy a perfectly suitable pair of tires for under or around US$50. See what your LBS has on offer.
Now, if you go roadracing, that's another matter... budget for $100 worth of tires (that's for two) every 1 000 miles or less. Or, ride for a tire manufacturer!