Before we begin, I'm a firm believer that for the most part you don't need a fancy $10,000 RED camera to get the good shots. I started taking pictures on an ancient HP point and shoot (circa 1998), then my iPhone, then a GoPro and now my 1DXM2. Start with what you can. You may find that you need that performance gear later on but don't think that you can't get good shots just because you don't have the top of the line stuff.
Buying your first DSLR is a bit of a jump from a phone or point-and-shoot. You've got new settings you've never seen before and buttons– so many buttons. Before you get too hung up on what exact model you want, check that it has the basics for what you need.
The very first thing I will recommend looking at above all else is; What is the sensor size? This will change the amount of light you can capture, and the viewing angle of your lenses. Once you have decided on a sensor size, you will want to consider the format, DSLR or Mirrorless.
DSLRs have a large mirror inside the body that reflects the light coming into the lens up into the viewfinder. When you press the shutter button the mirror flips up, the shutter opens, and light falls onto the sensor. This is what makes that deliciously satisfying shutter sound. DSLRs tend to have longer battery life, better auto-focus systems, and more variety in lenses available. In a mirrorless camera the light passes through the lens and right onto the image sensor and displays a preview on the rear screen (or EVF).
Mirrorless cameras have the advantage of generally being smaller, lighter, and faster than a traditional DSLR. If you're looking for a mirrorless camera my recommendation is the Sony a7sII; but it carries a pretty hefty price tag for a first camera.
ISO Sensitivity
ISO stands for International Standards Organization. But what it really means for you is how sensitive your image sensor is to incoming light. When you have very little light, say you're shooting stars or a dark night scene, you'll need higher sensitivity to capture more light. If it's a bright day and you're taking pictures in the snow with lots of reflected light, you'll want a lower ISO to reduce the sensitivity of the sensor to light coming in.
DSLRs and mirrorless have a much wider range of ISO than a point-and-shoot or your phone. This is why all those photos from that concert I took on my phone look like a grainy mess. When you can, user the lowest ISO you can while still exposing the shot enough. If you use too high of an ISO you will see significant graininess in your photos. If you're a beginner, you may opt to simply set your ISO to Auto and let the camera do the work for you.
Shutter Speed
Shutter speed is fairly straightforward. It's the amount of time that the shutter stays open and lets light in through the lens to activate the sensor. The longer the shutter stays open, the more light and the less sharp that moving objects appear. If you're shooting from a moving car, or taking sports photos you will want upwards of a 1/800th of a second. If you are shooting starry skies you will want multiple complete seconds of exposure time. Most cameras can stay open as long as you hold down the shutter (this setting is called BULB) or on the opposite end can be open for several thousands of a second. For comparison, my 1DXM2 will shoot as fast as 1/8000th of a second.
Aperture
This won't be specific to the camera body, but the lens that you purchase for your first DSLR (or mirrorless, whatever, I don't discriminate)
Every lens has an aperture range. Aperture adjustments are measured in units called f-stops. The higher the f-stop number, the more closed-down the aperture will be and the less light that passes through.
Higher f-stops increase the distance in front of and behind the focus point, reducing the depth of field (how blurry objects appear in the background are compared to the focused foreground). However, they decrease the light available for exposing the shot. Lower f-stops will give you a nice depth-of field affect and lots of light to expose your shot.
Smaller f-stop lenses (like a f/2.8) are sometimes called 'fast' lenses because you can take a photo at an extremely quick shutter speed and still properly expose the shot.
Finding a Model to Buy
One of the more popular beginner dslrs are the Canon Rebel line. They're APS-C sensors, and can usually do video at 1080p 30fps (what you would need for sufficiently good quality on facebook/youtube/instagram)
Most DSLRs are sold as a 'kit' which means that they come with the camera body and an 18-55mm lens which is a pretty broad zoom range for all around general shooting. 18-55mm is a good beginner lens but depending on your shooting style, you may find that you'd rather have a different lens. Some of these kits will be packaged with a ton of extras; You really don't need any of that and will likely want to buy things a-la-carte. Just check that a kit has the body, a lens or two, and maybe an extra battery. You can buy a camera as body-only but then you would need to buy a lens separately.
Lens hero is a great website for finding the lens you want. http://lenshero.com/
I find that the two types of lenses that I use the most are a variable zoom (24-70mm) and a wide prime lens but it really depends on what sort of things that you're going to shoot. If you're unsure of what to get I'd say start with a kit that comes with a good variable zoom lens like an 18-55mm, and consider picking up a portrait lens like a 50mm prime.
If you're looking for a good starter kit, many photographers will list their starter DSLR on 2nd hand marketplaces like Facebook, Craigslist, and Adorama for a pretty decent price once they are ready to upgrade.
This is a great example of a photographer getting ready to upgrade and selling gear on the Facebook marketplace for well below the cost of a new kit.
If you're dead-set on buying a new kit, these are some of my current recommendations for each of the most popular manufacturers.