Its gone through a soft reboot so nows a great time to get into it, theres just 1 short new series for the Doctor and its quite so-so but it has some good episodes, the new doctor and his companion hard carry them tbh.
Each doctor and their series are generally self contained adventures, there are some overarching storylines but the series will often remind you about old faces and storylines and make it easier to follow for newbies.
You're not missing too much if you start watching now, its sort of like how One Piece started the final saga and that sort of recap arc with Egghead too and this FMI arc remaster too.
That would be nice if you did! I strongly recommend watching all the doctor's from no 9 (christopher eccleston) to no 12 (peter capaldi), the day of the doctor movie and the 10th doctor's tv specials or w/e.
The series has amazing acting, writing, music and tons of heart, emotion, charm at times. There are some really amazing episodes out there!