If you want to get serious about photography, a DSLR offers significant advantages in image quality and control over a point & shoot. It also brings considerable bulk, complexity, and cost to the table as well. For people with no interest in photography that want nice pictures, I usually recommend a good point and shoot. For people that are interested in photography, I always recommend a DSLR.
The budget question is a significant one. You can get started with a DSLR package for well under $1,000. You can also spend well over $10,000 on a complete system. I usually recommend that you start with an entry level DSLR and focus on learning and acquiring better lenses.
All of the entry level DSLRs on the market today are quite good. They all have relative strengths and weaknesses, so you need to decide what is important to you.
Buying a DSLR is different from buying a point and shoot in that you are buying into a proprietary system of gear. Many accessories that you purchase (lenses, flashes, and other items like remote controls and batteries) will only work one particular brand of camera. There are also differences in the way that work (although the core functionality is all the same). The major brands on the market (Canon, Nikon, Sony, Pentax / Samsung, Olympus, and Sigma) all have enough gear available now to meet the needs of almost every generalist amateur. As you get into more specialized forms of photography, you'll still see significant differences between the brands. So even when you are buying an entry level DSLR, look at the overall ecosystem for your brand - what gear is available, what can I purchase locally, what is available for rental, who can I mooch gear off of, etc.
