Throwing gets all the glory, but catching wins games. The best handler in the world is useless if the person they throw to can not haul it in. And here is the thing: catching is a skill, not a talent. You can get dramatically better with focused practice.
Two Hands, Every Time
Use two hands whenever possible. The pancake catch (one hand on top, one on bottom, clapping the disc) is the safest, most reliable way to secure the disc. It works for throws at your chest, throws to your side, and even most throws above your head.
One-handed catches look cool. They are also how turnovers happen. Save the one-handers for when you have no other choice.
Reading the Disc
The disc is talking to you from the moment it leaves the thrower's hand. Is it flat? It is coming straight. Is it tilted inside-out? It is going to curve. Is it high with a lot of spin? It is going to hang and float.
Learning to read the disc in the first half second of its flight is what separates good catchers from great ones. You do not run to where the disc is. You run to where the disc is going to be.
Catching in Traffic
In a game, someone is always near you. The key is to attack the disc. Do not wait for it to come to you. Take a step or two toward the disc and meet it aggressively. This shortens the window for your defender to make a play on it.
The Layout
The full extension dive for a disc you can not quite reach. It is the most exciting play in Ultimate and every player should learn it. Start on soft grass. Practice diving from your knees first. Then from a slow jog. Build up to full speed layouts over weeks, not days. Your body needs to learn how to land safely.