I think I might be able to help here a bit, as I've stared using one recently. YouTube has been an awesome resource but here are a few things I picked up on my travels:
Apologies in advance, this ended up being a super long post. I've bolded the short answers to your questions:
I'd imagine at the very least a budget set up will be a huge a step up from rattle-cans. Obviously, they're not going to be as reliable or consistent as an Iwata or Badger, but the reviews should be able to tell you if they're workable. I've seen a few on Amazon where people posted photos of minis in their reviews, and I'd take that into consideration. There are a few things to keep an eye open for though:
Make sure it's a gravity feed brush.
If you can find one at a good price, get a dual action - you'll have control over both air and paint flow simultaneously. If you want to do anything else with this besides prime, it's worth the extra investment.
I went with a compressor that included a moisture trap. Not sure how optional this is - perhaps someone else can weigh in on that.
Moving on, yes you should absolutely thin your paints, but not the airbrush primer. I found the primer didn't 'grab' or cover very well when I did. I'm using this stuff. If you're using airbrush paints for models those should come pre-thinned.
I've been putting VA Model Color through there with their airbrush thinner and it's been fine, 1-2 coats usually does the job. (I'm still not entirely sure what the real-world difference between flow improver and airbrush thinner is.) Use a little more water than you think you need.
If it gets clogged or starts acting weird, I keep a wet sponge nearby for cleaning the tip. This also comes in handy for forcing a backflow to mix paint/rinse it out. If you see bubbles in the cup and you're not doing this, it's clogged. Dump & store the paint in a spare pot and flush it out with water.
For cleaning it out afterward, it depends. I'll defer to the manual. Backflow a lot with water to get all the paint out though. Finish up with whatever the manufacturer recommends you use as a solvent and blow air through to dry it completely before storing it. I usually take the nozzle apart and soak it in cleaner for a while too. Not every time but occasionally it'll be a good idea to field strip the thing and give it a good cleaning/lubing up. Careful: don't bend that needle!
Between colors, and depending on what I'm doing, sometimes I won't bother cleaning it. You can get absolutely gorgeous gradients and color-shifts this way. Otherwise just backflow with water.
Finally, a personal horror story: I was so excited to be able to finally prime minis indoors that I underestimated just how much overspray was involved. I duct-taped myself a big cardboard shroud/booth to work in and still ended up cleaning primer dust from every square inch of the room. The floor, upholstery, curtains, bookshelf, ugh. It was everywhere, meaning there was probably a good deal inside me too. Wear a dust mask - the technicolor boogers will cease and your lungs will thank you!
(I have since *mostly* avoided the mess by cutting the back out of the shroud and duct taping a furnace filter and a box/window fan behind it to suck air through. It's ugly as sin but works reasonably well. It's louder than the compressor though.)
There will be a learning curve but overall it's been a lot of fun and saves a boatload of time basecoating. Hope this was at least a little bit helpful, I'm no expert but I goof up a lot so feel free to ask me anything. Happy painting!