Ultimate is a very accessible game in that it requires very little equipment that you probably don't already have, aside from perhaps the disc. Here are some recommendations beyond that:
Shirt
Obviously, you should wear something comfortable and suitable for running and sweating in. For team identification during pick-up games, you should bring both a light and dark t-shirt.
Shoes
Again, comfort is key. You'll be doing a fair amount of running. In competitive play, players normally wear plastic cleats, and they are not uncommon during our pick-up games. So if you have plastic cleats (i.e. not metal baseball spikes), you are welcome to wear them, but they are not necessary.
Water
Your health is paramount, and nothing is more sorely missed during an Ultimate match than a drink of water. Be sure to bring enough water (or drink of choice) for yourself.
Disc
It never hurts to have your own disc, either for practicing on your own or for playing pick-up. We recommend getting a UPA-approved model. These include various Discraft, Wham-O, and Daredevil discs.