Perpendicular to the joists.

I would use construction adhesive and ring-shanked nails or screws to secure. If screws, make sure to drive so heads are not high. Try to hit joists.

I've read some advice not to hit the joists. And not the glue. It looks like there is some conflicting advice out on the internet...

Are you doing tiles or hardwood? If you doing hardwood you can do it as others said... if you doing tiles you should use something else like uncoupling membrane, etc

is it going to be a floating floor, possibly engineered hardwood, or vinyl

Overlap the existing seams glue and screw or use ring shank flooring nails and it will be fine. Be sure to re-nail it or screw down existing subfloor before you put down the underlayment.