Standard to compact swap

The front derailleur height always needs to be adjusted when making this switch. You should assume the front derailleur will need to be completely re-adjusted.

Dropping in a crank for a higher speed generation than the chain you're running usually results in some degree of rub between the chain and the large ring when in some of the small/small combinations. In some cases you might get chainsuck-type issues with the chain riding up on the ramps in these combinations too. The problem can typically be solved with chainring spacers. I do a set of Wheels Mfg 0.6mm spacers preemptively and it usually works.

Hypothetically, going from a 14t gap to 16t you could exceed your rear derailleur total capacity. In practice, the max cog and total capacity numbers of road RDs for some time now have been set up to avoid this possibility.

If your bike has a Shimano external bearing bottom bracket, a newer Shimano crank will still fit. Shimano has kept the crank spindle diameter and length the same.

12-28 cassettes were common with 53-39 cranks. If this is what you have be aware that you will have a lower top gear ratio. You can change to an 11-28 cassette and get a slightly higher ratio.

If changing both the cassette and cranks you will need to replace the chain as well.