It's the sort of question that lights up the eyes of salespeople and makes professional photographers twitch: "Which digital camera should I buy?"
Let me make a couple of points up front that should be obvious: (1) While I keep up with many trends in the wider photography world, my personal experience is limited, and (2) I am unquestionably biased in my recommendations. So take specific model suggestions with a grain of salt; the major players all sell roughly comparable models at various price points, and you can take fine photos with any brand of camera. Don't let anyone try to tell you that brand X will make you a better photographer. As the famed photographer Ansel Adams said decades ago: “The single most important component of a camera is the twelve inches behind it.”
With those caveats, I'll offer some considerations for the person who is looking for a camera that can offer more creative control and capabilities than their smartphone can deliver. Usually, when people ask for camera recommendations they have a budget limit in mind. (If you're rich, skip this part.) If your budget is under $300, your options are limited and I'd suggest sticking with your smartphone; you're not going to get an appreciably better photo with a camera in this price range. A cheap point-and-shoot camera offers a few advantages, but frequently not enough to justify the cost. (If you're looking for an optical zoom and a more powerful built-in flash, then you might actually benefit from a sub $300 camera.)
If you can spend in the $300-400 range and up, your options are much better. Even if you have the means to spend a lot more money, I'd recommend that you consider a more basic camera kit until you have a better idea what kinds of features you really need. With a few high-end exceptions*, you'll be better served by a camera that will accept interchangeable lenses. These cameras usually come with a basic "kit" zoom lens that are more than adequate for everyday use. If you shop carefully, you can find some good deals on refurbished gear at great prices - like a Nikon D3100 with an 18-55mm lens for $349 at the time of this writing. That's more than enough camera to take professional quality photos. Nikon's 18-55mm kit lens is quite good and surprisingly versatile.
As your skills improve, you can add lenses that match your shooting style. As any pro photographer will tell you, your most important investment is lenses. Even the best digital cameras today will start to look ancient beyond 5 years as technology progresses. But a quality lens will last you through multiple camera bodies. So if you need to scrimp, buy the cheapest body a manufacturer offers and save up for better lenses.
Don't be suckered into the megapixels arms race! As Ken Rockwell demonstrates, megapixels are not the most important spec in a camera. Even a 5+ year old DSLR can take photos that beat any smartphone in image quality under normal circumstances. Among other factors, dedicated cameras generally have much larger image sensors. What do you get if you cram more pixels into a tiny sensor? A lot of pixels on a tiny sensor! For making beautiful 8x10" prints, a dedicated 6 MP camera is all you need. More megapixels do allow you to make much larger prints, and you can also make bigger enlargements from cropped images. Today you're not likely to find a new camera that features less than 16 MP, so don't worry about megapixels. Megapixels simply aren't a measurement of quality - only resolution. Of course there are advantages to newer and higher-end cameras, but megapixels are not the chief among them.
I've owned a number of digital cameras, including a Nikon D40 and a Nikon D7000. Both my DSLRs served me well, and would have continued to do so if I'd stayed within the Nikon fold. About a year ago, however, I decided I wanted to change up my gear and go a different direction. I spent considerable time researching mirrorless (also know as ILC - interchangeable lens) cameras. These are high-end cameras that produce photos rivaling the quality of conventional DSLRs. The main difference is that they don't use a mirror like a conventional DSLR for composing photos. Consequently, they are generally much smaller and lighter, and - to the delight of many professionals - are less conspicuous in public. Another advantage of the mirrorless cameras is that you can often buy inexpensive adapters that will let you use vintage lenses on them. I can use my 30+ year old Konica lenses in this way, with excellent results.
Mirrorless cameras do have a few disadvantages compared to DSLRs. Their smaller size can take some time to adjust to, although this is a minor adaptation. Due to their smaller batteries, battery life is measured in the hundreds, not thousands of shots like my former Nikon D7000. Newer cameras are improving on this limitation, but fortunately spare batteries are cheap and widely available. The biggest drawback for mirrorless cameras relates to slightly slower focus speeds in some models (the actual difference is increasingly negligible), and fewer frames per second. Wildlife and sports photographers are probably still better served by a DSLR. For most any other purpose, mirrorless cameras are functionally equivalent to their bigger brethren.
There are a variety of options within the mirrorless camp, and this is not a comprehensive review of them. I decided upon the Fujifilm X-E2 mirrorless camera, which at the time made the most economic sense for me. The Fujifilm X series digital cameras offer the advantage of a using an APS-C (half-frame) sensor - the same sensor in the Nikon D7000 and many other "prosumer" level cameras, with some special Fujifilm features. Fujifilm's XF lenses, even their entry-level "kit" 18-55mm zoom, are widely recognized as exceptional in quality. The primary disadvantage of going with Fujifilm is that there are relatively fewer lenses to choose from, and only a few lenses made by third-party manufacturers.
From the various photography forums and podcasts I follow, it appears that mirrorless cameras are gaining significant traction in the market, whereas DSLRs sales and innovation have mostly stagnated. Some of the most exciting developments are coming not from Nikon or Canon, but from manufacturers like Sony, Panasonic, Olympus and other manufacturers of micro four thirds cameras. These are mirrorless cameras that offer many advantages in common with the Fujifilm X series, with the added benefit that a vast selection of lenses (interchangeable among brands) are available.
I almost bought a micro four thirds Panasonic Lumix camera, as it has some useful features and has garnered numerous positive reviews. Ultimately, I decided against going with micro four thirds out of concerns that the smaller image sensor used in micro four thirds cameras might not provide the performance I required - especially under low-light conditions. My concerns may or may not have been well-founded in that respect, but I'm definitely happy with my camera choice. When I decide to splurge for an upgraded body, the X-E2 will become my backup body, and my current lenses will work on either body.
*If you're looking to buy a good camera that uses a built-in lens, Fujifilm makes some great consumer cameras that are worth checking out; two of these are the Fujifilm X30 ($599), and the Fujifilm X100T ($1299). Both are only available for pre-order at the time of this writing, but early reviews suggest they are every bit as stellar as their predecessors. The X30 is a compact zoom, whereas the X100T is a fixed lens camera that's going to be very popular among professional and street photographers. If you don't care about having the very latest features, the X100S is predictably being discounted by retailers. The X30 does have a smaller sensor, whereas the X100T has the latest iteration of same excellent sensor as other cameras in Fujifilm's high-end models.