Certain types of bassline go with certain types of music. Country, for example, is very much root-5th dominated (and is handy for other music forms where you haven't quite worked out what to do but want to add a little variation).
"Saw her standing there" uses a variant on a walking bass line, which generally takes the form root-3rd-5th and then either 3rd-root or 6th-flat 7th and back down. Someone has written a treatise on the walking bassline with all the rules about what you can and can't do, and it gets quite mind-boggling, so I just stick with the simple variants. Anyway, that will serve for most blues songs. Note: Folsom Prison Blues is a country song, I've heard it played with the bassist (who is a guitarist who owns a bass) doing a walking bass line and it sounds utterly shit.
The basic pentatonic square is a useful device - root, 4th, 5th, flat 7th, octave. Dropping the 9th in after the octave as a passing note can be effective too.
Transition from major to relative minor (eg C to Am) - drop in the 7th as a passing note, so C B A in that case.
The more strings you have, the more chances you have to look flash by playing the same thing in different places on the fretboard. Sliding up an octave and then doing a fill on the pentatonic square is guaranteed to impress (as long as you hit the octave).