At our house there are two things that save the day all the time during change-the-diaper time that we had no idea we would need before we had the baby.
The first are waterproof changing pads that we put on her changing table over the terry cloth cover. When she (occasionally) has an accident mid-change, it more often than not takes the whole mess. Cleanup is way easier - just wipe it up and sanitize.
The second is way more simple but equally important: paper towels. Get a roll and keep it handy in the baby's room, because when the baby pees all over the waterproof pad, it's going to take more than baby wipes to clean it up. And keep reminding yourself that urine is sterile. Poop? Not so much.