Basically you want to learn on a car that isn't your main car. And it needs to have something wrong with it that you can fix. It doesn't make sense to take apart a working car just so that you can mess with it. Buy a used car with a fixable problem, and fix it yourself.
My recommendation would be to find a 4-cylinder car on Craigslist that has a blown head gasket. You can probably find an old Saturn for $500 or less. Do the head gasket repair yourself. In the process you will find out all about removing/replacing all the external engine accessories (alternator, air conditioning compressor, water pump, power steering, etc.), the internal engine timing (timing chain or timing belt), intake and exhaust manifolds with associated sensors and controls (throttle body, O2 sensors, TPS, EGR, IAC valve), and how to torque stuff correctly. You will probably spend about $50 in parts and need to buy about $200-$300 in tools before it's all said and done. Buy the Haynes and/or Chilton book for the model car and use it to help you. There are also all kinds of helpful videos on YouTube, depending on the popularity of the car in question.... there is also probably a dedicated forum on the web somewhere for just about every make and model of car where you can ask for help.
I recommend a Saturn because I recently replaced the cylinder head on one myself, having never done such a thing before. There are lots of Saturn fans out there, and Saturns are fairly easy to work on, and all the old 4-cylinder models (SL, SC, or SW) have pretty much the same 1.9-liter engine so it's fairly easy to find info. Then when you are done you can either keep the car or sell it and probably get your money back, including what you put into tools.
I would not recommend getting a car or truck with a "V" engine (V-6, V-8) to work on for your first project. They usually do not leave much room to work on the engine while it's still in the car, and pulling the engine is probably asking too much for your first try at this. Something with 4 cylinders will usually be roomy enough in the engine bay to work on without pulling the engine.
A head gasket is a good, fairly easy problem to fix for a first-timer. It lets you dig deep into the engine, but there's really not that much that you can screw up. Things like fixing transmission problems or bearing replacement is not for the beginner IMO. Replacing the clutch might be an option, but detailed transmission or engine rebuilding work is for the pros IMO.... unless you've got a LOT of specialized tools and time on your hands!