If you're new to CrossFit you may be a bit confused about footwear and apparel (Deadlift day means we seeeveryone in leggings, meggings or long socks) and you will see many flat soled and minimal shoes or even lifting in socks. Maybe lifters. You will not see running shoes or those with lots of cushioning.
Why? Because if your shoe has cushion, it affects the way you squat, the way you receive the weight or distribute the weight, ultimately making the way we need and cue you to move, harder.
Here's a pretty great deal from Reebok. Nanos are a shoe that will allow you to lift, squat, rope climb, jump, run, and double under in. It's the best bang for your buck.
While you need nothing special to CrossFit, we'd encourage two things: get a flat soles shoe and your own jump rope.
If you want to get fancy and like all types of gear, you may want a pair of lifters. These are made to be able to do a metcon in but it's best to use a lifter only during a heavy lift or something that you need the most stabilization in (like a heavier OHS for many).