Personally, I wouldn't waste my money on cheap compact cameras.
Go for DSLR, even if it's the cheapest one. Even entry level DSLRs are very very good
What's more important, it's the set of accessories, like lenses, flashes, diffusors etc etc.
I recommend Nikon or Canon (Canon will probably be cheaper in the long run). These are 2 big dogs in DSLR market, there are others, like Sony, Pentax - but are practically non existent for the professional, so why bother.
Once you buy a camera, you will have to stick with the same brand for the accessories (unless you go for third party lenses/flashes). But Nikon and Canon have basically the same stuff.
Buy any DSLR + 18-55 kitlens + 50mm 1.8 lens + an external flash with cheap wireless trigger + cheap tripod.
Add some patience and you can get very good, dare I say professional, results.
I did semi-pro photography, before my kit was stolen