The first thing that you'll need to do after you have the water off is to make sure that you know what is wrong with the sink sprayer you're going to fix. There are a few things that could be wrong. First, a lot of sink sprayers have reduced water flow because there is something blocking the front of the sprayer.
One of the most common problems like this is hard mineral deposits. You should be able to tell if this is the case right away - there will be a hard, white residue all over the sprayer head. This is also very easy to fix. Just take the sprayer head off and clean it. The best way to clean these is just to soak them in a vinegar bath. The slight acid will eat through the deposits.
Sometimes you will find a sink sprayer with mineral deposits that just does not get clean in a mineral bath alone. In this situation, don't worry! It just means that you have to do a little bit more work, but you will get it cleaned up in the end. A toothbrush is a good way to clean off the deposits without harming the actual sprayer head. Don't forget to clean inside of the head as well! There are probably plenty of mineral deposits inside of the sprayer in addition to the ones that are outside of it.
Finally, if the sprayer is still not working correctly, you may need to tighten or replace the diverter valve. This is a valve located underneath the faucet itself. The purpose of this valve is to put water through the sprayer while it's in use instead of through the faucet. In order to repair this part, you'll need to rip up the entire faucet.