I think it’s a good idea to have CPR and First Aid training. If something should happen, it’s good to know how to take control of the situation. I had a player sprain an ankle in one session, so I was able to wrap it correctly with an ice pack and bandage until his dad came back to pick him up. Plus, these things are good to know anyway, even if you aren’t going to coach. And you never know when you might need to use CPR skills.
As for the injuries that might happen, most parents and players know what kinds of risks are involved with sports. What you have to watch out for is any kind of horseplay on breaks when you are doing small groups. If you just do single-player lessons, this isn’t an issue. And even in the small group setting, the kids who are coming to the lessons are there to learn and are usually not the type to fool around.
As far as getting insurance, just do a search on Google for “liability insurance private sports coach” or something similar. There are companies who offer policies. Here are a couple links that I’ve come across:
Another thing that I’ve come across is parents dropping off their kid(s) for the lessons and then coming back to pick them up at the end of the hour. I’ve never had a problem with that. The longer you do this, the more you get to know the parents and players, but you never really can be 100% sure that you’ll never be accused of something.
One idea to cover yourself is to have a video camera or your phone camera record your sessions in their entirety. Just set the camera on the sideline or in a corner so it captures everything for the entire time of your training session. Like I said, I’ve never had a problem with something serious like this and I've never heard anything like that from the other coaches I know, because most parents will hang around and watch you do the training. But you have to decide for yourself what you want to do if you have parents drop their kids off for lessons.
At the very least, if you film the entire lesson, you'll have plenty of stuff to review with your athlete, which you should already be doing anyway.