What would be smarter, getting budget headphones and an amp, or getting more expensive headphones and no amp?

Answer 1

An amplifier can't improve the sound of cheap headphones.
so, of the two options, the second one seems better.

Answer 2

Assuming you mean better quality when you say more expensive the headphones would be much more important. Even a relatively inexpensive amp can do a very good job, but you can't make crappy headphones sound better.

Answer 3

Get good headphones that don't need an amp.