Here is the beautiful thing about Ultimate: you do not need much. A disc, some cleats, and an open field. That is it. No expensive equipment, no special facilities, no gear bag that weighs more than you do.
The Disc
The official disc for Ultimate is the Discraft UltraStar, 175 grams. This is the disc used in every sanctioned game from middle school leagues to the professional AUDL. Get one. Feel it. Learn how it sits in your hand.
If you are brand new or playing with kids, start with a softer disc. The UltraStar Soft is made of rubbery silicone. It flies like the real thing but does not sting your hands on a cold day. It is also nearly indestructible, which is a plus when you are learning and the disc is hitting concrete more than hands.
Cleats
Soccer cleats or football cleats both work. You want something with good traction that lets you cut and stop hard without slipping. Molded cleats are safest for pickup games. Metal cleats are not allowed in most leagues.
The most important thing? They need to fit well. You are going to be sprinting, stopping, pivoting, and cutting for an hour or more. Blisters end more Ultimate careers than injuries do.
What to Wear
Shorts and a shirt. Light colors in summer, layers in winter. Bring a dark shirt and a light shirt to every pickup game so you can play on either team. Gloves are optional but helpful in cold or wet weather. Layout gloves protect your hands when you dive for a disc.
Water and Sunscreen
This sounds obvious, but I have watched grown adults play two hours in July sun without water. Do not be that person. Bring a full water bottle. Wear sunscreen. Reapply at halftime. Your future self will thank you.