What do I need to know when replacing my inner tube/tires?
Basically you just need a tube that's the right size (diameter and width) for the tire.  And of course you need the right stem -- the thin Presta stem, or the thick auto-style Schraeder stem -- depending on what your rim is drilled for.

There are various grades of tubes, but you most certainly want just the standard grade.  Thin ones designed for racing leak down rapidly, while thick ones designed to resist punctures are really bulky and hard to deal with.

Be wary of a shop that doesn't have the right size tube (the range of sizes should be printed on the box the tube comes in) and tries to sell you what they have in stock.  While a "slightly wrong" size will often work, it will be hard to install and will be apt to not run smoothly.

[Also note that for a small cut in a tire you can often get away with using a "boot" behind the hole.  This can be a specially made piece of thin, flexible plastic, a piece cut from the sidewall of a lightweight tire, or even a piece of folded currency, if yours (as in the US) is fairly rugged.]