Converting to 3x from 1x
Frames sold with 1x builds may be designed in a way that's freed of the constraints of FD bikes.

They might not have the small or large ring clearance needing for any given multi-ring crank.
The seat tube might not accommodate a front derailleur by virtue of its diameter, placement/angle/curviness, whether it's even tubular, etc. If it's round and in the normal place, but a nonstandard diameter that's less than 34.9mm, you can typically solve the problem by just beer can shimming up to 34.9 and using a 34.9mm FD. There are also some braze-on FD adapter mounts that have been made for odd diameter seat tubes, but that wouldn't help for non-road bikes.
There may be no accommodations for FD cable routing, although in most cases if you've got a seat tube that can take an FD anyway then you can solve this problem fine using a clamp-on housing stop and a top or dual pull FD, then run full housing to that stop and zip tie it to the top tube.

A pretty common scenario from having done this on a few different bikes is the first two aren't issues, but a clamp-on housing stop is needed on the seat tube. It makes sense from a manufacturer's standpoint to offer the cleaner look and not have unused housing stops, so that's what most of them are doing. I've also seen one 1x native bike I can recall with a seat tube diameter in the 33s.
Many bikes, especially hybrid/urban bikes, will do exactly what you propose without issue. You may need a new bottom bracket depending on the crank you pick out and whatever you have now. The chain will need to be replaced if it was sized properly to begin with and now you're going to a larger chainring. You'll also need to make sure your rear derailleur has the total capacity to handle the new gearing; bikes that come with 11-34 and an basic long cage MTB/hybrid derailleur will tend to have total capacity for a triple, but bikes that had wide-range cassettes to start with may not.