Hello Oobee35. Are you planning on keeping female mice or males? I would advise three or four females if you are looking to have a small group. Male mice are notorious for fighting and usually end up separated after one has been badly attacked or bullied, but if you keep a single male, he will require more human attention but not too much handling, just assurances allowed to walk onto your arm.
Single males can become very tame but patience is always the key with all mice.
If your in the US, I`m not familiar with the cages available over there, but don`t overlook the Zoozone 1 Perspex tank-style cages for mice even though they only have barred lids and the lid would require meshing to prevent escapes. There are other cages with much narrower bar spacing. I`m sure there was a cage called the Simba or Simbla? Can`t remember as it`s been a long time since I was looking at cages!
Mice are indeed good climbers but don`t necessarily require a lot of height in a cage. A one storey cage with a platform or hammocks etc....will be fine and even transforming a tank-style cage with a build in platform would work. Keep it as natural to the mice as possible. Their urine does smell, especially male mice, so spot cleaning is the best way to maintain their environment without using scented sprays or anything that could affect their breathing because mice (like rats) can suffer terribly from respiratory ailments.
Any barred cage with 1cm or less bar spacing is the way to go. Anything wider could cause problems. I wouldn`t spend money on a cage you may end up regretting because it looks nice but isn`t practical for the mice.
Look at what is available and come back here and ask opinion on them. Will be happy to help.