There have been many Star Trek movies as well as TV series and I didn't watch any of them. So, do I have to watch complete or any Star Trek TV and movie series to watch Star Trek Beyond or just the last two movies (Star Trek and Star Trek into Darkness) in order to understand plot points?
In the end it is really up to you. To understand and enjoy the story, all you need are the recent two movies (I'd say).
If you want to understand and notice parallels, nods to the original series, etc. you'd have to watch that as well (including the movies).
But besides that, they did a pretty good job to establish characters, setting etc. without forcing the audience to know everything from past installments.
The only direct canonical reference to the old series is essentially the overall setting (reintroduced) and old Spock in the first Star Trek reboot movie from 2009, but that's it. The Original Series, The Next Generation etc. they all didn't happen in this timeline (except Enterprise).
Star Trek (2009) is kind of a reboot. It takes place in an alternate reality to the previous films and TV series. There are nods to previous films; like old Spock etc. but they are self explained in the film.
You can still watch the old stuff but they are not compulsory to watch for understanding Star Trek Beyond.