Lady Jemima, welcome back!
Dual action airbrushes seem more difficult to use than they really are. Kind of like patting your head and rubbing your tummy at the same time, it takes a while to get the hang of it. Once you "click" with it, it will become second nature.
Small compressors get hot real fast. They aren't designed for use over an extended period of time. The key is to take a break and let it cool down.
How to clean is dependant on what media you are using, but the differences aren't that great. Cleaning the needle is pretty straight forward, using your finger is not a bad method, but be careful! The needles are, well needles and are very sharp.
The tip can be cleaned with a Qtip and whatever thinner is appropriate for the media you use. If it's acrylics simple ammonia works the best (the blue Windex is a good choice). Careful to pull back the needle tip so it doesn't get damaged.
I use only acrylics so I squirt Windex through my brush at fairly high pressure (I use a garage style compressor) and follow up with distilled water. I use distilled as I'm worried about mineral buildup as I live in a hard water area. Then I blast air through to dry it out. I remove the needle to insure it's clean and wipe needle lube on the front half so it moves nicely. The needle lube also prevents "dry tip".
If you're using enamals and lacquers I'll let somebody who uses them answer how best to clean! It probably won't be much different. Likely only the cleaning fluid will be different.
edited for tiep0$!