After I spent some time at the beach during vacation, I got sun spots on my foot and lips, should I check them out?

Yes. Sometime benign pigmented spots called labial macules can appear after sun but sometimes these can be a sign of a more worrisome condition that can sometimes be cancerous. The best advice is to have these spots evaluated by a board certified dermatologist. Also recommended is a lip balm with at least an spf of 30 applied to the lips on a daily basis.
Yes, as a rule. Great question! - yes, any new "spots" on sun exposed - or even non sun-exposed skin should definitely be screened for skin cancer -with a dermatologist - and may even need to be biopsied. -> even more important if you have a family history or personal history of melanoma. Good luck!
Check new spots. If one follows the golden rules, than it might improve the diagnosis of skin cancers so any new lesion that doesn't go away in a few weeks should be checked by a board-certified dermatologist. Pre-existing lesions should be checked if there is a new color, shape, border, if it becomes asymmetric, or grows, or itches for a few weeks or bleeds.