The idea that DSLR's have better image quality is pretty realistic. In general, they have larger sensors, and sensor size rules when it comes to image quality. There are however a lot of other factors (Lenses etc), and not all DSLRS are better then non-DSLRS. 


The best thing about buying into DSLR's (or any other large sensor, interchangeable lens cameras) is that the best camera's you can buy (without taking out an second mortgage) ARE DSLR's (including their full frame mirror-less counterparts), and that a cheap DSLR is starting you on the right track. Mirrorless options are also good to look at, but in general (if buying new) their seems to be less "cheap" mirror-less stuff, then cheap DSLR's. Also note, if you go into an retailer, they are going to push Mirrorless onto you, as there is a lot more margin (and incentives) in them then DSLR's


I'd suggest buying the cheapest body you can, something like an Canon 1100/1200D (or Nikon/Sony equivalent) which you can pick up sub $400 second hand with a lens, then as soon as you can afford it, upgrade the lens. Canon make a great 17-55mm f2.8, there are also third party companies like Sigma that make great budget lenses (Sigma 18-50 f2.8).


At the end of the day, although gear does make a difference, just getting out there and shooting with whatever you do have, will make the most difference.