When I’ve had to get rid of just one or two trees (a few inches wide), I cut them off a few feet above the ground, dig around them, cut through the roots (one way or another) and use the trunk to start rocking it back and forth. Eventually you can get it out of the hole and not be left with a stump.
Sounds like you have a bigger problem. You could call someone with a stump grinder to get rid of them. The other option I’ve always heard is to drill a bunch of holes in it, fill it with gas, wait a day and fill it with gas a again and continue to do that for a few days. Then you’re supposed to set it on fire and let it smolder for a few days to burn out the roots. That always made me a bit nervous, but everything I’ve read says that even if you don’t set it on fire, all that gas is enough to make sure it’s totally dead. There’s also other chemicals you can buy to soak the stump with using the same method.
If you do this, eventually the stump and roots will decompose/decay and disappear. The idea is that they’ll do this without leaving you with a yard full of mushrooms.