Not sure what all your looking to test, but these are some good things to have:
soldering kit:iron/flux/solder(i like having a number of different gauges around)/and a solder pad for wiping the iron on are the essentials.
isopropyl alcohol is great for cleaning electronics, apply to a rag and wipe. make sure the power is turned off obviously
a good rag. depending on areas you need to get into a shammy or glass cleaning rag are good choices, generally go with something like a glass cleaning rag because it will be thinner/allow more detailed cleaning. The benefits of these over a normal towel/rag/paper is that they wont leave behind little pieces of paper or cotton.
a good jewelers screwdriver set. these are made to fit your smaller sets of screws/spaces.
volt meter. dont go super low end, it wont have all the necessary test functions. look for something like this
http://toptronic.com/product_info.php?cPath=23&products_id=43&osCsid=0spsv2av1ncecjdj38qkgffqvlo5u5haas far as screws/bolts/washers go you can either buy a kit with a lot of types or simply pick up new ones at a hardware store from measurement. little side note here: you can always buy a box/case/small drawer based storage system and save all your old small screws from broken electronics. ie before tossing that broken xbox take it fully apart and keep all the screws, sorted for size.
canned air can sometimes be useful, but i rarely use it.
on the fan issue, if wiping it down doesnt get it up to full speed then either the bearings are going, or the grease is too gunked up. while repairable the cost of fans being what they are it would be better to just replace.
anyway with those items you should be able to fix a rather decent portion of hardware issues. after that its about replacing burnt out chips/capacitors etc. which you would need a voltmeter and a soldering kit for(which you may or may not own) use the voltmeter to find the dead part(s) and replace with new if possible. One last note on the voltmeter these are very useful for testing correct voltages are getting to your parts/their arnt spikes or drops in voltage from say the PSU.
One other note:if you choose to take apart your PSU to clean it be VERY careful. Even unplugged it can store enough amps to kill you. cleaning its fans is one thing, but cleaning the other parts you should make sure you know what your doing before hand. for the inexperienced i would recommend isoproyl for the fans, and canned air for the rest. it wont get it spotless but you dont have to worry about stopping your heart either.