How do I give 3D-printed parts in PLA a shiny smooth finish? The surfaces of my printed parts using PLA plastic look rough and uneven.

Would changing filament to a better one make any difference?

If not, what kind of methods can I use to achieve a smoother finish for my for 3D-printed objects?
PLA parts can be finished with a coat of epoxy like XTC-3D from Smooth-On. This will smooth out the part and give it a pretty nice shine.

I've also had a fair amount of success sanding prints, giving them a coat of automotive filler primer, and using glossy spray paint.

You can also get great results with an acetone vapor finish if you're willing to switch to ABS. Though that will require a heated bed and can be a bit more finicky to work with than PLA.