I certainly hope it's not bad to ride horses on grass! That's all we have around here. But no, I'm almost 100% sure it's not bad for them; besides, of course, slipping or something, like Silvera said. If the ground is soft it might be bad for the grass, because they can tear it up.
Riding wise, grass is about the best thing to ride on, but as ilyTango mentioned, if it's wet, it can get a little slippery and if very wet, it doesn't take long to get churned into mud.
Also, I am not sure if I should board or keep at home.
The decision is really about how big a part of your life do you want your horses to be. If you want to be able to go to the barn, ride, socialize with friends, and then be free to leave, boarding is for you. If you enjoy having your horse with you 24 hours a day, caring for them, and have a place to ride, home is the place.
Our mares are at home with us and I can't imagine having it any other way. I like to go out, spend time with them, and saddle up and ride whenever I feel like and having to drive to a barn just wouldn't be any fun at all (and it takes time and gas). I do occaisionally trailer to the ranch down the road where some of my friends board to ride with them, but to be honest, I don't miss the barn drama that seems to crop up at all the barns.