I second everyone who said don’t use poison. Mice don’t sit around and eat till they’re full. They carry it away and hide it. Thus, the possibility that food in your pantry will be contaminated, or that you’ll find bait pellets in closets and slippers and such does exist. Also, the line about them getting thirsty and going outside to die while they look for water is a crock. No one can guarantee that. They may die before they get outside, or they may become too sick to move before they get outside. If they die in some inaccessible part of the house, they’ll stink for a week, and you’ll just have to live with it. The pest control companies who suggested poison were very irresponsible, in my professional opinion.
Glue boards work well, as sometimes you get multiple catches on the large ones. If you dislike the idea of them possibly mutilating themselves or dying a lingering death, then don’t use them. Snap traps are quick and painless, but multiple catches are out, and you have to make sure you check them daily. Put them where you see evidence and in corners (mice huddle with their backs to a corner when they eat, so they know nothing is sneaking up behind them). Place the device perpendicular to the wall, with the trigger closest to the wall. That way they will catch a mouse approaching from either direction.
Do an inspection, also. Since mice can fit through a hole no larger than a dime, it’s impossible to find every access point, but there may be a place near a door or a pipe or cable where something has deteriorated and created a space large enough for them to get in. Then seal it. Start your inspection near where you see the most evidence, and don’t neglect crawl spaces and garages. Those are also prime spots for traps. Peanut butter works well as an attractact, as does a cotton ball soaked in vanilla extract (don’t know why, but it works).
Keep in mind that mice are good climbers, and can access a hole higher up than you might give them credit for. It wouldn’t be a bad idea to seal pipe openings under your sinks and in the laundry, too.
Don’t worry about them reproducing faster than you can catch them. At the beginning, you’ll probably catch adult mice. Then, as time goes on, you’ll catch smaller adolescents who are forced to forage further to find their food, as the adults aren’t around to bring food back to the nest. Any babies won’t be able to sustain themselves after that. And, if you seal as many access points as you can find, you may eliminate future problems