Welcome to the world of digital photography. Most of us, like you, have come from the simple point and shoot cameras and realized early on that this was something that was fun and rewarding. Back in the day, we shot film cameras which were slow and hard to deal with. The film had to be developed and we would wait days for our exposures to be processed. We took prodigious notes just to remember what adjustments were applied during an "exposure" so we could replicate the shot later on. Still, many of us hung in there and with the advent if "free" digital pictures and instant results, really realized how bad it had been. I have several shoe boxes filled with photographs from this era. At about 20 cents apiece, I've vested thousands of dollars in these almost "keepers".
Your first DSLR (Digital Single Lens Reflex) camera is a major decision. There are two major players out there, Nikon and Canon. If you use windows on your PC, you probably like to have more control of your environ; I suggest Nikon. If on the other hand you favor the Mac operating system, I would sway you towards the Canon system. They are not compatible, ( Nikon lens will not work on a Canon and vise-versa) so you should weigh your decision carefully. Whatever you decide, remember, its expensive and difficult to switch (lenses and gear are expensive and not interchangeable).
Start with a simple DSLR and a curious nature to find your niche; whether that being portraits, landscapes, astral, people. places or things, they are all rewarding. I suggest looking at the Nikon D3200, D5100, or the Canon EOS Rebel series t3i or t4i they are all equally qualified to open the doors for your interest. Get involved with this site, as there are many people who contribute their expertise in the many areas that digital photography envelopes. Keep it real simple and read the users manual. Don't get discouraged or frustrated-if you have a problem understanding something just ask. Soon you will be helping others who, like you, have decided to take up this rewarding and enjoyable hobby.