What should one do if they arrive late to davening and they wish to catch up on what they have missed. If the minyan is currently saying Shemoneh Esreh- what should that person do in order to make up the davening they missed?

The question is too broad for a short answer. Get a good siddur (the various Art Scroll siddurim have excellent instructions) that has the rules for coming late. In general, one may skip sections of, and sometimes all of pesukei d'zimra in order to catch the minyan for shmoneh esrai, but not Kriat Shma or it's brachot. If you come in to Ma'ariv when the cong. is beginning SE you can doven SE with them and then say Kriat Shma.