When a nail cracks or splits, nail glue or mending tape is your best solution. Dot the nail glue onto a piece of a tea bag, or coffee filter paper, and cover the nail where it is torn. File the nail smooth, and cover it with polish if you like; this should last until you can get to the salon.
When a tip falls off, you can glue it back yourself with nail glue, but be sure to make an appointment with your nail technician so that it can be filled in, and you can avoid mildew from forming beneath the nail.