I have experiences on "both sides".
One thing is to just hang around the places band is sure to visit (tour bus is a great place to hang if you can get around it), and be
patient.
But nowadays it's just a lot easier, having a family with musician who is in quite popular band, so we wives and
realgirlfriends (not those who you find tangled up your sleeve and begging you to take her backstage) usually get free
passes to shows and festivals.
Because of that, I've had a chance to meet my idols and many many bands I've never would've even dreamed of meeting!
And I can tell you, it is one of those WTF- is this real?- moments when you realize, that THAT world-wide famous finnish
band is having a barbeque party at OUR backyard with our friends

(they had a gig here, so they needed something relaxing before it)
My best advice:
Hang around places where the band is passing/ doing stuff
Advice #2:
If you see someone carrying something, stay out of their way

Advice #3:
Act properly. Don't annoy anyone.