There are several sizes of stump grinding machines, from smaller units that you can attach to your back, and bring slopes or hills, and grind medium to small size tree stumps. And then there's a medium-sized tree stump grinder that you can usually rent from your local yard shop.
Then there are bigger size machines, which need more space, and usually, fill more, but can do it faster. You also have a giant size machine that can get you going, if you can find clips on YouTube or cable television.
We will assume you will do it yourself, and you rent a standard 13hp size stump grinding machine. The front of the machine has a blade almost like a circular saw, but with a carbide tooth. These teeth are very hard, and if you want to grind it yourself, you need a green grinding wheel to do it.
The second most challenging thing is small stones. If your stump is on the ground that has small stones, you should try and dig as much as possible before you start. You want to install a wall of plywood around your work area, so there are no small stones or flakes that fly, especially if you work around windows or fine objects.
Most cities have telephone numbers that you can call if you want to dig in your yard. Most grinding stumps usually go down about 6 inches below the class, so you can insert concrete, soil, new grass, mulch, a new yard or whatever, and the engine can go deeper than that, but deeper, harder to control the engine.
After you have the engine in front of the stump, set the bar above the stump, and then make sure the wheels are released from the stump, and start the engine, each engine will be different, some will be pushed themselves, some with brakes, some with different things. One of the stump grinding machines that I have is to go close to the house or fence and bring it to the side, without hitting objects.