Reddit mentions of Vittoria Zaffiro Pro Home Trainer Fold Tire

You don't need specific tires, especially starting out. If you're committed to a training program and don't want to flatten out your good tire, a trainer tire like this is something to consider if you've got a spare rear wheel lying around but it's certainly not a necessity.

With a trainer, you lock the rear wheel into the trainer and make contact between your tire and some sort of cylinder that drives a flywheel which provides resistance. With rollers, you put your bike on top of three cylinders and keep it there through balance, patience, attention, and sheer force of will. If you're asking about the difference between rollers and trainers, you don't need rollers. Most training programs take for granted that you're using a trainer and not rollers - especially ones that focus on your power zones (rollers typically don't have any resistance on them - SportCrafters being a notable exception). Rollers are also harder to learn how to use properly and typically significantly more expensive than trainers of similar quality.

If you want a suggestion, go with the CycleOps Magneto trainer - it's about $250, pretty solid, easy to set up, provides progressive resistance (no knobs or levers to fiddle with) and is relatively quiet. If you can find one on Craigslist or something, so much the better. After a couple years you may want to upgrade and you'll probably have a better idea of what you're looking for based on your personal preferences.

Thanks for the quick response and the education. So I think this should work:

• Wheel - $48 - http://www.bicycleoutfittersindy.com/product_p/we8677.htm (depth is 16mm, but as you said, brakes are not a concern for trainer)
  
• Cassette - $38 - http://www.amazon.com/Shimano-CS-4600-10-Speed-Bicycle-Cassette/dp/B0065FIE52
  
• Tire - $25 Vittoria-Zaffiro-Home-Trainer

Tube + Trainer tire. Be advised should you purchase the trainer tire. It's a real tight fit. The Amazon page has a nice exploded view.

https://www.amazon.com/Vittoria-Zaffiro-Home-Trainer-Fold/dp/B007IEHWDO