I wholeheartedly agree with the call for having your tetanus vaccination up to date. Spasming on the floor with lockjaw so bad you can't breath you wouldn't be much good to yourself or anyone else.
Second, you can use a big bandage on a small wound but a small bandage is nearly useless on a major wound. Band-aids are fine to have and they save time, effort and bulk but make sure you have large surgical bandages and/or trauma wraps in sufficient numbers. For those who don't know: Sanitary napkins make good bandages for heavy bleeding and large wounds. And they are both cheap and pretty much sterile.
Third, A pack of "steri-strips" wound closures can be used as-is for small lacerations. Duct tape can also be used but keep a bit of gauze between the skin where it is cut. Otherwise there is a god chance of a reaction to the rubber used as adhesive on the duct tape.
You can also use super glue. What get missed sometimes is that those strips can be used with super glue and can largely replace stitches and staples in all but the toughest situations. These strips, with or without super glue, or duct tape can be used to reinforce stitches or staples in spots like elbows or knees where lacerations tend to reopen.
As counter-intuitive as it may sound the old adage about a sharp knife being safer is very true. Your less likely to need to use the sort of brute force that invites an accident if the knife is sharp. If you do slip, everyone does eventually, a sharp knife produces a cleaner and less painful wound. I have seen a deep cut from a very sharp knife close up and heal in a very short time. A cut from a dull knife tears and destroys a good bit of flesh. Devitalized tissue is a prime spot for an infection to set in.
I'm glad it all worked out. Eventually everyone makes a mistake and minor cuts are the inevitable result. The only people who have never cut themselves are those who don't do anything. It pays to be as a safe as reasonable precautions will allow but eventually everyone will slice themselves. It is part of the game.
I have a friend who did her research in deep jungles with remote tribes. She described toddlers who played with a razor sharp machetes about as big as they were. She was alarmed. The women told her not to worry. The child is too weak to pick up the machete so while they get cut the kids learn very quickly. After a few minor cuts they learn how to handle this vital tool with respect but without fear. She noted that most adults had a few scars from childhood. These were seen as badges of experience.