Airbrushes are one of those tools that it's worth spending money on upfront as opposed to buying cheap, then better later. You're original experience will dictate whether you'll ever use it well, so chasing your tail with a subpar piece of equipment isn't going to help, only frustrate.

You're better off skimping on the compressor if need be, but a good brush is a must in my opinion. My main brush is an Iwata HP-BC2 that I bought over 30 years ago. It's all original, even the needle and still works flawlessly. Last year I bought another HP, this one gravity feed so I can change color frequently for figures and the design and quality hasn't changed one bit. The HP line is their professional grade, my original was well over $300 back in the 80s, you can get them around $200 now, but their hobbyist brushes are outstanding too. You can easily stay in your budget with one, I suggest checking out Chicago Airbrush, great prices and service.

In terms of compressor, you just need air at a steady rate, it could be an inner tube if need be. I'm using a 30 year old Badger, but any simple machine from HF or Home Depot will do fine. Regulators and moisture traps can be added afterwards if necessary to fit your budget. The more expensive AB compressors get, it's usually for quietness. That's a big concern depending on conditions. My spray booth is on the third floor above bedrooms and I spray at all hours, so quiet is important. If you're set up in a garage and will only spray at reasonable hours, it's different, a tank compressor that will work with other things may be a better investment (run tools, inflate tires, etc).

For Airbrushing models, you generally stay in the 12-30 PSI range (a good regulator is a must) so almost anything out there will produce that.