Tweezers for getting splinters out. Invest in a pair with a narrow point and solid grip. If the tweezers don't work, it's best not to dig the splinter out with a needle because that's likely to lead to infection. The splinter will work itself out eventually, usually overnight. Both are cheap and great at killing germs but should be reserved for cleaning up dirty cuts and scrapes, since they are powerful enough to damage healthy tissue.
Peroxide stings a little more than alcohol, which feels cool.
An antibiotic ointment such as bacitracin to dab onto cuts and scrapes to ward off infection while they heal. For fresh cuts and scrapes, wash with a mild soap first. For cuts that have some dirt in them, clean with soap and water, and wipe with half-strength hydrogen peroxide. Then dab on a little ointment. Unless the wound is serious, it'll heal faster if you leave it uncovered.