I just started rock climbing recently (by the way, it's awesome!). I tend to tear up my fingertips though when climbing on rough rock or for long periods of time. It takes a couple days for them to heal enough to where there's no pain when I press on them (after one day, the pain is minimal, but they re-open easily). Is there any way to help them heal faster?
Listening to your body is spot on. Let the finger heal completely before climbing again. This is very important, as a split tip is very likely to split in the same spot again.
Prevention is very important, and will help keep tips from splitting in the future. Start by moisturizing the skin regularly, especially after climbing. Use products like Climb On and Joshua Tree Salve: they will help the skin regenerate. This is very important in the winter when the skin is drier.
You can use an emory board to sand away dead skin; smooth skin is less likely to have a line of weakness that can split. If you have a big callus that is about to rip, file it down.
Your skin will build up calluses, but it does take time to adapt. Don't rush it, and keep things progressive. Pick climbs that don't have extremely sharp holds. When climbing, make sure your hands stay dry: use chalk to prevent soft skin that can split.
I've heard people using super glue to heal splits, but I've also heard it's a terrible idea from other people. Tape can also be used over the split, but I find the loss of sensitivity very distracting.
Not really. Not the answer you want, but your body is telling you something, you need to listen to it.
It's a lot of fun, but you need to take it slow to ease your body into it. Otherwise the fun is going to come to an end.
I've heard a lot of stories from people that buy things like these: http://shop.nicros.com/index.php/warrior-boardtm.html then go crazy trying as many one knuckle pullups they can before their muscles and tendons are strong enough for it, and ultimately injure themselves so bad they have to take several months or a year off of climbing. It sounds like you're on your way to doing something similar.