Here are the questions I think you need to ask yourself before buying any new camera.
What's my budget?
The amount of money you can spend on camera gear will probably be the biggest limitation on getting any specific camera. It will sway your decision on whether or not you would prefer getting a fixed-lens camera, rather than an interchangeable lens system camera. With fixed lens cameras, like your Coolpix, the spending pretty much stops there. You buy the camera, maybe some extra batteries and cards, a few accessories, and you're done spending. Interchangeable lens cameras, like dSLR and mirrorless cameras are quite a bit more expensive, because the camera body is only half the camera--the other half are lenses. And the system also can include lighting and support gear, as well as bags, filters, etc. If you are thinking of getting a dSLR, consider that what you spend on the body is liable to be only a third of what you need to spend to round out a basic system. Think in terms of US$1500-$2000 starting with a $500 camera body.
Your budget will typically determine which models you'll be able to look at.
How important are convenience vs. image quality to me?
Today, while dSLRs remain the largest system and most versatile of the digital camera types, there are also fixed-lens advanced compacts and mirrorless cameras that can easily equal dSLRs for image quality, low light capability, and exposure control, while being quite a bit more convenient/smaller/lighter. How heavy a bag are you willing to lug around all day when you're taking pictures? How often do you wish you could just put a camera in your pocket and go? You aren't limited, as we mostly were five or six years ago, to just P&S pocketable cameras and dSLRs. There's a huge range of cameras in-between that you may want to consider as well for a best fit. And what will fit you best depends not just on budget, but also on usage.
Try to figure out whether what you want/need is a dSLR, a mirrorless system camera, or an advanced compact camera (with or without a large sensor).
See also: What do I need to consider to choose between dSLR, mirrorless, or a compact as my first "serious" camera?
How do I plan to take pictures?
What are your most-like picture taking scenarios? What kind of shooting do you know you want to do? If you're just starting out and you don't have an idea, then sometimes a dSLR is a great choice, because it has the most options on ways to go. But if you know your primary area of shooting will be travel, or street shooting, or while camping, then possibly a smaller lighter camera has the priority. If you envision using a tripod and cable release to do landscapes that you're going to print large, then maybe a full frame dSLR is what you're after. Do you plan to shoot portrait photography in a studio? Sports on a playing field? Wildlife? Or are you chasing the rugrats around the kitchen? Some cameras are better than others for fast-action photography; some systems have better supertelephoto options. Figure out how and what you plan to shoot or concentrate on shooting the most, and other more experienced shooters can tell you what the preferred gear might be.
"Better" is often in the eye of the beholder. Which brand, model, and lenses you might want mostly come down to what you plan to use the gear for. All the cameras today are good, there really aren't any bad ones--it's a matter of where the relative strengths or features of a particular camera model can meet your specific needs, so be clear about what those needs are.
Am I sure it's the camera and not me?
Some folks go into a new camera purchase with the unrealistic expectation of the camera making better photos. A camera is still just a tool. You still have to learn how to use it, to get the most out of it, and to see and compose a good, strong image. Make sure that it's really the camera that's holding you back, rather than technique or lack of knowledge. Maybe you don't need a new camera as much as a tripod, a flash and lighting know how, a book, a class, a workshop, post-processing software and know how, or just 10,000 hours more practice with what you've got.
From Aaron Johnson's What the Duck 174: