The most important thing to remember is that whatever notes are marked sharp or flat in the key signature will be sharp or flat through the entire piece, no matter what register you play them in.
So how do you figure out the name of a key signature? Easy! To find the name of a major key signature with sharps in it, take the last sharp (use the order of sharps to help you) and go up a half step. Example: a key signature has two sharps, F-sharp and C-sharp. One half step up from C-sharp is D, so you are in D major.
To find the name of a major key signature with flats, the second-to-last flat in the key signature is the name. Example: a key signature has three flats, B-flat, E-flat, and A-flat. E-flat is the second-to-last, so you are in E-flat major. Use the order of flats to help you.
Finding the name of the relative minor of a major key (this is the minor scale that uses the same key signature as a particular major key), uses the exact same process for both sharp and flat keys.
Take the name of the major key signature you're working with and go down two letter names on the staff (not counting the one you started on). The note you end up on is the name of the minor key, and if that note is sharp or flat in the major key signature, then that sharp or flat is included in the minor key name.
The best way to get comfortable with finding key signature names is to practice! When you play a piece, see if you can find the major key name and its relative minor. Here's a key signature cheat sheet for your reference. Print one out and keep it in your band/lesson folder!