Well, the next logical place would be The Next Generation. It was a great series, and just about all other series branch off of that. The first season they were trying to figure things out, but after that it was great.
There is also the old original Trek cartoon series which was ok, but you can skip it if you like.
From TNG, go with Deep Space 9, that one was excellent.
Voyager came next, but honestly that is the only Star Trek series I never bothered to finish. Maybe watch a couple seasons to at least get to know 7 Of 9, as she shows up in the Picard series.
Enterprise was good the first couple seasons. This is a prequel set before the Federation was founded. It went downhill for me in the third season with the whole Xindi storyline that took up the rest of the series.
Discovery was interesting, especially since the main character is NOT the captain. Some people didn't care for it, but I liked it. It is set before the original series, and even includes Captain Pike for a season. (captain of the Enterprise before Kirk)
Spin off from that is Strange New Worlds, which covers Captain Pike in command of the Enterprise. I'm liking it so far.
Lower Decks is a great animated series about the crew that do all the work on the ship rather than the command crew. It is filled with easter eggs and references to the other series. Lots of fun, but might want to wait until you've seen TNG, DS9, and Voyager to get most of the references
Prodigy is another great animated series. In this case some young slaves in a mine find a lost Federation ship. The ship has a program of Captain Janeway meant to help train new Starfleet Recruits, and takes the escaped slaves as starfleet recruits to train. They escape in the ship, the person running the mine was specifically mining the planet to find that ship and starts hunting them down. This one does not require knowing about the other series to enjoy, so you can watch it at your leisure.
Picard just finished its final season. Have to have watched TNG to enjoy this one, and it includes 7 Of 9 from Voyager, so should watch those others before this one.