Don't waste your budget on things like rash guards and shorts. A pair of sports shorts and a tshirt will do. I got a pair of board shorts for £10 from a local clothes outlet and they are basically the same as any sprawl shorts. They are even made of this funny(like neoprene) material that has loads of give in them and secure ties around the waist.
Concentrate on specialist essential items. Mouth guard, open hand grappling gloves, 14/16oz gloves for sparring and if your budget allows, shin guards and headgear. Those are not essential though, especially when you're just starting out. You'll probably be spending more time on pads and bags than people.
Here are the shorts I was talking about. Can't say enough good things.