Point and shoot camera's can make nice photos, no doubt about it.
If you're planning to grow a lot though you're probably going to want some sort of DSLR.
The entry-level DSLRs aren't too expensive (at least not when you compare them to the big boys

).
You could look into something like the Nikon D3100, D3200 or D5100. They're very capable cameras and not too expensive.
Canon, Sony, Pentax and some other brands also make very nice entry-level cameras, but I don't have much knowledge in those areas.
Just go to a store in your neighborhood and hold a few different cameras to see which you like best.
Also do a bit of research in lenses, you're going to be needing those.
Remember that once you buy a camera brand you'll be needing lenses that fit the camera. Canon lenses don't go on Nikons for example.
There are some 3rd parties like Sigma and Tamron that make lenses that fit on the big brands. They're cheaper usually (a bit less in quality sometimes but they still make very nice lenses).