@rikku couple of things going on here of which one the water damage will become one of your main issues. First off skip the rice. It is absolutely useless. It will not help you fix your phone. All it does it provides a false sense of security. It will not clean your phone, it will not prevent corrosion.
Backup all your files right now. Don't do anything else but that. After you've backed up your files stop continuing to try to charge, sync or otherwise use your phone. This potentially can make a bad situation worse.
The very first step I would take is to disassemble your phone. Use these guides to disassemble your phone. Take a careful and close look at all components and connectors. Truly assess the damage. Look for major corrosion, evaluate the pins in the connectors etc,. Take a look at all connectors as well as cable ends. After that I would suggest that you clean it a couple of times in sterile water, not drinking water. Use a soft brush and clean it very, very thouroughly with the sterile water. You will have to remove all the EMI shields. Once you got the gross contamination cleaned, use 90%+ isopropyl alcohol and clean your parts some more. For a proper cleaning, use this guide. It was written for a Apple iPhone 3G but it is still pertinent to your phone as well. I can not stress the importance of a good cleaning enough, so do it over and over while replacing the alcohol after each cleaning. Do not use compressed air for the cleaning since that can drive liquid as well as corrosion particles into areas that will cause trouble later on. Do not use drinking water, if you need to use water, use sterile water. Also, you do not need to leave it to dry for a day. Isopropyl alcohol in the higher concentration will evaporate quickly and dissipate the water. The importance here is the cleaning. If you can get it professionally cleaned via an ultrasonic cleaner, go that route. Do not forget to replace the battery. This is a must and not an option. It will fail if you do not take care of it now. Keep in mind that water damage is the hardest to troubleshoot and to repair.
As for the "glitches" on your screen, that looks like a failing displaying from short-circuiting by the water. You will have to replace that as well.