I’m so sorry to hear that you have a mouse problem. Count yourself fortunate that it is limited to the detached garage. Let’s start with preventative measures.
Not only do you want to prevent damage of belongings, but you need to discover what the draw is. It could just be shelter, but there is most likely something that keeps them around. In my house, it was an open tin of dog biscuits on a shelf in the garage. In Mom’s, it was a sack of bird seed. She now stores it in airtight plastic containers.
Secondly, look for ways of the mice to enter the garage. From what I’ve read, their bone structure is fairly flexible, allowing them to work their way in through small crevaces. Another article on the internet suggests putting vinegar-soaked cotton balls around entrances. Apparently, mice are repelled by the smell. Personally, it sounds like a lot of work to keep this up, but if it works, so be it.
It also sounds as if it is time to have a major garage cleaning and reorganization. Even if there is nothing inside to attract the mice other than shelter, know that they poop like mad. Better to clean and declutter now than later. Their droppings become a health hazard to the family.
The traditional mouse traps are probably the most humane and one of the cheaper solutions for ridding an area like an infested garage. As for what to use as bait, I’ve heard cheese, peanut butter, jam, etc. Just try different items and see what works. They are hungry little critters and will eventually seek out what is offered.