As Ray stated, you need the EF version of the lens for your 70D. And the 70-200 f2.8 is a very versatile sports lens and is on one of my camera bodies at every sports event.
With the APS-C sensor, the comparative focal length is 112-320mm (with respect to using it on a full frame body) which means you need to be fairly close to the action if you want a lot of detail for individual players. The 70D low light/higher ISO performance results in loss of detail so you don't want to have to crop too much for late afternoon or evening games.
I use full frame cameras, typically with a 400 f2.8 on one body and a 70-200 f2.8 on the other, and even though I am on the sideline the 400 typically gets used for over 60% of the photos in a typical soccer or football game so you will want to get sidelines access if you can, if not you will often find the visitor seating is a little closer to the playing field than the home side at many stadiums.
For volleyball, the 70-200 f2.8 is the perfect focal length for your camera when shooting from near the gym floor. Volleyball is very much a timing specific sport for photographers and you cannot rely just upon shooting high speed bursts to get the action you want given the speed of the game and ball movement. My 1DX III will shoot sustained bursts at 16 FPS but for good volleyball action images, precision timing works far better than the "spray and pray" burst method. The learning curve is steep but short so don't get discouraged if you aren't happy with your first volleyball photos. I had shot a lot of sports before the first time I shot volleyball and after the first set I felt like the dumb kid in class taking a multiple choice test because I felt I could have done just as well closing my eyes and shooting at random. For the next set, I put both cameras in single shot mode and concentrated on anticipating and timing which produced far better results for me. I now shoot volleyball with the cameras in high speed mode but most of the images I take are single or at most bursts of three captures; shooting a set in single shot mode taught me a lot about the discipline of shooting volleyball. For me, it is still the most difficult sport to shoot and definitely not my favorite.
"...shooting mostly my kids and grandkids sports, soccer, volleyball and football, etc. Is this a good lens for that?"
The correct answer is, yes and no! As stated above the most important thing by far is where you shoot from. However, most of us have to make do with what is presented at hand.
For football and soccer the 70-200mm is probably too short especially true from the stands. I only have shot one volleyball tournament in my career. I took a variety of lenses but found the 50mm f1.2L and 85mm f1.2L were the best choices. I was granted access to wherever I needed to be as long as I didn't interfere with play. A big positive.
I use a 150-600mm Sigma Sport for football on my 1D Mk IV and 1DX. I use a 120-300mm f2.8 Sigma Sport for basketball on my 1DX.
The second most important thing in shooting sports is to know the sport. If you have no idea of what is going on you will almost invariably be out of position and not ready for the shot.
Another issue might be your shooting technique. Using longer FL length lenses brings some additional challenges.
One last suggestion is to get a good post editor. Every great photo you see and love from the pros goes through some post editing. You can get a good one, DPP4, form Canon for free. Even mediocre shots can be made better with it.
One last, last suggestion with no additional charge is to always shoot Raw format.