Steve,
One of the best things you can do for an old airplane is help prevent corrosion. Keep the airplane clean, waxed and ready to go. Particularly the underbelly and gear areas. By regular cleaning you will become intimately familiar with your aircraft and see smoking or popped rivets, loose or lost screws, cracked items and you can immediately drill a hole to stop the cracks and possibly avoid a costly repair. This is nothing more than getting on a car dolly and cleaning with airplane safe cleaners. You can even do a bit of touch up painting from a spray can or airbrush paint touch up to keep things looking smart. This item alone can help save you maintenance particularly if your airplane is on a tie down and subject to more wx damage. Obviously this does not require mechanic aptitude.
Then oil changes, cleaning plugs, inspection of all hoses fuel lines, scat/sceet hoses, exhaust system for wear/leaks. WE inspect these things before every flight but I mean a thorough check from three angles from one end to the other of every single hose.
You can check for exhaust leaks by putting the exhaust side of a vacuum to blow into the exhaust and use some dish-washing liquid in water with a small brush and dab it around all connections to see if you can find any soap bubbles just like checking for a leak on gas lines in the house or a/c lines. To actually fix a leak you will need supervision but just to try and find the leak you should not.
You can change brakes, tubes and tires without supervision. I'm not sure you might even be able to repack the bearings as well. You just need to have a set of airplane jacks and have someone show you how to use them safely.
I have heard guys say they pull their own failed accessory: alternator, starter, carburetor or mags to send out to be IRAN'ed when needed then get supervision or an A&P to put the item back on the aircraft and make the log entries. (some pilots have told me they replace the accessory never make an entry themselves of course this is not legal and I am not recommending this). Maybe not the starter if it requires removing the prop. I would not want to take the prop off myself.
In some aircraft dropping the exhaust and lines to take off a cracked or failed cylinder is a rater easy thing to do. This is more than my adventure level but I have helped my mechanic all six cylinders on my current airplane.
I know others who have removed their jugs to have them Nickle Nitrated to avoid rust on aircraft that sits weeks between flights.
There is a tail horn inspection AD on Comanches and many of the ICS guys remove the tail and horn to be inspected and then have the mechanic help replace it. Shaves a few hours off an 8 hour AD.
If you want to do an owner assisted Annual you can at very least take off all the inspection plates, remove the seats from the airplane, clean the battery box and repaint it, clean plugs, change oil, change tires, install new battery, help with compression check, you can pull all your Ad's from FAA directly and if you want to go above a normal annual you can even find the Apache SBs and FAA problem serviceability reports for areas that are found to need more attention.
But like everyone says, take a weeks vacation and do the next complete annual with your IA and you will see many opportunities to perform future maintenance if you have the time and interest and a willing mechanic.
Just wondering how many of you started out who take care of your own planes.
I do my own work on my cars, but I've not really done anything on my own aircraft. Did most of you just find a local A&P who'd inspect and sign off of your own work and start off with basic things, or did you take an official class/course?
Running wires and turning wrenches doesn't bother me, but I want to make sure I do everything nice and legal.
) I've assisted painting, fliberglass repair, wing replacement, reskinned wings, welded on airframes. 
