Get an old bike and repair it.  You can find all the information on line if you search around.  There are probably books that cover soup to nuts if you want a more thorough introduction.  I recently repaired two bikes replacing inner tubes, cleaning and adjusting the brakes, chain, shifters and replacing a three piece bottom bracket bearing with a new cartridge...the latter a job I had not done before.  I considered replacing the chain on one bike but took a see how it goes approach because I'm a cheapskate. 

Bike hardware is generally fairly standard to a particular era and style of bike.  For some jobs you need or at least it is helpful to have a specialized tool that goes with that era...I had to get a crank remover and cup wrench to replace that bottom bracket...but generally you can get away with a few screwdrivers, an adjustable wrench, hex keys and perhaps a couple of tire "irons".   A chain removal tool can be useful once you go beyond adjusting the brake and changer cables.