|Should have checked those brakes...|
What you need:Allen Keys or a Multi tool
Tire pressure gauge (unless the pump has one).
Shock Pump if you have air shocks.
If possible and not included with the multitool, chain break tool and spoke wrench.
First the easy stuff:Clean the bike. Take time to hand wash the bike, use the rags and soapy water (and de-greaser as needed) to carefully clean all the gunk and grime left over from last season. Don't use high pressure water, this will just wash the grease out of the moving parts (we don't want that). Once the bike is clean and dry give it a careful inspection looking for cracks that may have gone un-noticed under the dirt and grime.
Check your tire pressure. Top up your tires to the appropriate pressure for your riding. If you ride roads, crank em up to the max listed on the sidewall of the tire. If you ride mud, go with a number that works for you. Either way, after a winter of sitting, your tires are probably not going to be where you want them.
Now to the still pretty easy stuff:De-grease and re-lube your chain. Use the degreaser and the rags to get your chain nice and clean. If it's particularly bad, consider removing it and soaking it in a small bucket of de-greaser for a few hours. Once the chain is completely clean and dry, apply a generous dose of chain lube, wiping off the excess with a clean rag. While you're at it apply fresh lube to all the moving parts you can see.
Check shock air pressures as needed: If you have air shocks on your bike, use your shock pump to check the air pressure in the shocks and adjust as needed. Like your tires, the shock can bleed a bit of air over the off season. Or you might have gained/lost weight and need a new setting.
Check your shifting. With the bike either on a stand or while holding up the back tire turn the pedals and work trough all the gears. If shifts are not going cleanly into gear, adjust the derailleur as needed.
Now on to the slightly less easy but still pretty easy stuff:Check your brakes. If you have rim brakes, check the pads for wear and debris. Use tweezers or a nail to remove any small bits of stuff stuck in the pad. Make sure the pads are not too worn, if so they should be replaced. If they appear to be glazed over, give them a light buff with a file to prolong their life. If you use disk brakes, check the pads as well to ensure they are not worn thin, and if needed replace them. Check and adjust the brake leaver travel to suit your preference. In the case of cable pull brakes, you may just need to loosen the nut holding the cable to the brakes, pull the cable tighter, then tighten the nut. For hydraulic systems, you may need to add more fluid to the system. See the manual for your brakes or your local bike shop for proper procedures for this.
Check your rims to ensure they are true. Give the wheel a spin and look for bends. If you are comfortable working your own rim, use a spoke wrench to tighten and loosen spokes as needed to bring the rim back into line. If you are not comfortable with this have your local shop take care of this for you. While checking the rims, also make sure no spokes are loose. Grasp each spoke pair and give them a light squeeze and if any are loose use a spoke wrench to snug them up. Just be careful not to go too tight as this can pull the rim out of true.