What do I need to know when replacing my inner tube/tires? Context:

The inner tube on my bike burst the other day, I took it home and salvaged another one from a bike that's been sitting unused in my garage for years.

The New(old) innertube inflates fine but when I inflated it I noticed that the inner tube was spilling out through a gash in the stitching of the tire.

I've temporarily replaced the tire with another that I had lying about but the ripped one was a semi slick and the one I've replaced it with is a mountain biking one, I'd prefer a semi slick tire.

I know I'm going to have to replace the tire as well.

The new inner tube seems spongier and I can't seem to get it to inflate as much as the old one.

I'm a relatively new to (serious) cycling, and this is the first time I've had to replace an inner tube or a wheel.

I mainly cycle on roads and cycle paths, the cycle path I use it just rough enough for me not to justify getting a full on road bike.

Question

What do I need to know in order to go into a shop and get the right replacement inner tube/ tire for my bike? I'm aware that there are different sizes of tyre (mine is 26") and there is another number which I assume is width (Old was 1.5"). Is this all I need to know?

Do I need to know anything about the inner tubes? I assume the size is the same as the tire, are there different grades or types of inner tube? Why would one be spongier than the other?
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.]