"An additional question... If I have a 3 sprocket cassette in the front, and a 7 sprocket one in the back... with 2 shifters then, this bike should theoretically have 21 different "speeds"... right? So far I can only access 7 of them because the front cassette won't shift up gears... I can use the shifter and hold it in, which lets the chain jump to the middle cog however, when I let go of the shifter the metal bracket of the deraileur rubs the chain and forces it back to the small cog."
LIBERALLY flush the shifter with a solvent like WD-40. "Open up" the shifter as much as you can, but don't try too much. A watchmaker has problems trying to repair these!
Let it set for an hour and repeat. After another few minutes-
SLOWLY move the shifter as far as you can and
SLOWLY EASE it back down.
IF you are lucky, the hardened grease will have dissolved enough for the pawls to catch the 2nd position. Go from there for the 3rd position. (start at 2 and ease up to 3)
My main bike is an 86 RockHopper, so yours is a bunch newer than mine!
BTW the front "cogs" are called chain rings or just rings.
A good reference for various adjustments/minor repairs-
http://www.parktool.com/repair/