I don't think there's a general trend that we need to known regarding which elements can form polyprotic acids/bases. However, I think the rather we should just be familiar with the common ones, some being:
As for the second part of your quesion, you can simply look at the anion. In order to be a polyprotic acid, the acid must be able to donate more than one H+ protons. Thus, thinking logically, you can look at the anion of the acid to see whether it is polyprotic or not. For example, looking at a sulfate anion, it has a negative 2 charge, which means that it could potentially hold (and donate) 2 H+. However, when you look at a chloride anion with a negative one charge, it is only possible to attach one H+ and therefore cannot be a polyprotic acid. Looking at this process in the reverse manner, we can identify which bases are polyprotic. The anions of polyprotic acids are the most common polyprotic bases. Ultimately, polyprotic acids and bases can be identified by looking at the collective negative charge of the anion, which must have a negative value greater than 1. Sorry if this confusing, but I hope this helps!