If the tire is mounted wrong, you might work on that a bit more. It is good practice.

Are you able to mount (and dismount?) the tire with just your fingers?

I always do the valve first, and work away from the valve, but FBinNY's method is reasonable, with doing the valve last, then using it to clear any rubber under the tire. I'm just careful that I don't roll the tube under the tire, especially near the end.

The bike shop should be able to tighten the cones. If you bought the bike new there, they might do it for free
Watch the guys there if possible.

Here in Eugene, we also have a local bike co-op where the "mechanics" will help teach you about the bikes. See if you have a local one.