What is the best DSLR starter camera for taking action and still shots of animals?

I'm overwhelmed by all of the options. This will be my first real camera. I've been using my iPhone for pictures and am just tired of being so limited.

I mostly take pics of dogs playing and doing agility courses, but I want a camera that will also be great for doggy portraits, regular wildlife, and landscape shots.

Price isn't a big concern. I want to make sure I start off right!

Any ideas of what would be best for a beginner like me?

The problem you will have is not which DSLR camera to choose. It's the lenses you use with a DSLR that are crucial for capturing certain subjects. Like an ultra wide angle zoom, for landscapes. A good fast prime lens for portraits. Or a reasonably fast telephoto zoom lens for wildlife.

With DSLR cameras, there's not one lens that can do it all. That's why DSLRs have interchangeable lenses.

A good telephoto zoom lens for wildlife will be the biggest expense, probably several times the cost of your camera - it could stretch to thousands of dollars for such a lens.

As a beginner, I suggest you get a camera certainly, any entry level DSLR will do - but start off slowly, learn how to use it, and build up your lens collection gradually as your skills and abilities increase. The camera body itself is the least important part of a photographers kit. Skill and quality of lenses trump the camera body every time.

Oh, and join a photography club, or take a class - if you want to get there faster.

For the most part, I totally agree with B K. The body doesn't come into play except when you're doing sports or wildlife. When you're shooting moving subject (sports) you'll need a camera that can track the movement of the subject. Most any DSLR can do it, but not all do it well and some don't it well especially under low light such as indoor arenas.
You may want to skip the entry-level body in which ever brand you choose, and go one or two steps up. You'll end up getting a better and faster AF system along with faster continuous shooting. Entry-level bodies typically will only have about 3 frames per second (fps), but mid-level ones will be between 5-7. In order to maintain that speed, the AF system needs to fast as well, which is why in single AF cameras will have one speed, but in servo or continuous, the fps drops.
I wouldn't recommend a mirrorless camera like a Sony A7 series due to the ergonomics of the camera. It just feels awkward with heavy lenses such as telephoto zooms, and their AF systems are not nearly as good as those in similarly-priced DSLRs.

Bodies that you may want to consider:
Canon: 60D, 70D, (at $1,800 the 7d Mark II is probably the best DSLR for sports and wildlife under $3,000).
Nikon: D7100, D610, D750
Pentax: K3 Pentax has a unique feature not found in Canon or Nikon. They have image stabilization built into the body which provide IS with any lens that will fit it. With Canon & Nikon, the stabilization system is in their lens. There are numerous prime lenses by Canon & Nikon that don't have IS (Nikon calls it vibration reduction or VR), but with the Pentax you can have it with any lens which allows for 4 stops slower shutter speed when hand held.