Say I already had a Pokémon gifted to me from a parent or something, could I still go to a professor and get a “starter” from them? Is there a limit? Like what if I got a “starter” from kanto and then went to Hoenn. Could I get a “starter”‘there too?

This tends to depend on the regional league’s rules and the distributor’s discretion. In less populous regions where there are fewer trainers starting out (such as Hoenn), you may be allowed to obtain a starter from a distribution point, even if you already have a pokémon registered to you. On the other hand, the leagues of more populous regions with far more new trainers per year (such as Unova) may specify that the first pokémon you have registered to you, regardless of its origins, is your starter, barring you from obtaining a starter from a professor if you’ve obtained a pokémon from someone else first.

On the other hand, distributors may also either create their own rules or file for special cases. For example, Professor Sycamore of Kalos has, on at least one occasion, given an additional starter to specific trainers as part of his research, thereby suspending Kalos League rules for those special cases only. Conversely, while there are no starter restrictions in Alola, Professor Kukui and Kahuna Hala limit trainers who begin their journeys with them to just one starter, due to the specifics of Alolan tradition. (It’s thought that any pokémon given to Alolan trainers is their chosen partner, so those who are already partnered with a pokémon prior to the starter ceremony must undergo the ceremony with that pokémon, rather than one of the traditionally offered trio.)

Thus, it’s a good idea to check with both your local league and the person from whom you plan to get your starter, just to understand the rules in your particular circumstance.