Can I work on my own airplane without formal mechanic's training?
FAR Part 43 contains the information of what you can and cannot do to an aircraft and what requirements you must have. 

If you are a certificated Private Pilot (as in not a sport pilot), there are quite a few things you can do to an aircraft you own, that is not used for commercial purposes. Like:


Replace wheel bearings
Remove/repair/reinstall wheels
Replace safety wiring and cotter keys
Service landing gear struts by adding oil/air
Lubrication not requiring disassembly of non-structural items like cover plates, cowlings, and fairings
Simple fabric patches not requiring rib stitching or removing structural parts or control surfaces
Replenishing hydraulic fluid
Refinishing decorative coating of fuselage and other parts
Repairing upholstery without disassembly
Simple repairs to fairings, etc
Replacing side windows
Replacing safety belts
Replacing seats or seat parts with approved replacements
Troubleshooting and repairing broken circuits in landing light wing circuits
Replacing bulbs, reflectors, and lenses of position and landing lights
Replacing wheels and skis where no weight and balance computation is required
Replacing cowling not requiring removing the propeller or disconnecting flight controls
Replacing or cleaning spark plugs and setting gap clearance
Replacing any hose connection except hydraulic connections
Replacing prefabricated fuel lines
Cleaning or replacing fuel and oil strainers or filter elements
Replacing and servicing batteries


And others Source

However, note that it has to be your airplane, and it can't be used commercially (which I think includes offering it for rent). There is very little you can do in the way of avionics (read #31 and #32 in the reference I gave). 

The fun thing to note is that although a PPL is required, the PPL training does not give any special attention to any of these items, and therefore really somebody off the street is technically just as qualified as a private pilot, but the private pilot is the only one authorized to do the work (provided they own the plane). When I got my PPL, nobody walked me through replacing cotter pins or anti-back-out devices...

Edit
These rules though may be different when you get into experimental category aircraft but I can't source it now and will do more research and edit when I can clarify.