Trapping should be your first choice for two reasons. Each of these devices works differently, but you'll have better results with any trap if you follow a couple of general rules. Don't count on doing the job with just a single trap of any kind. Instead, get several and use them in different locations for a period of at least 15 consecutive days. If you're not catching critters after this time period, you can be fairly sure they're gone.
Snap traps are the most commonly used household traps.
All are designed to kill the rodent quickly and humanely. Snap traps are, well, a snap to use, and can be reused indefinitely if you choose. Glue boards are a safer alternative to snap traps when kids and pets are present. They're available in sizes suitable for mice and rats, although rats have been known to escape from these sticky beds. Some glue traps come prebaited, but it's easy to bait them yourself with a sprinkling of birdseed.