Deep Space 9

It's not quite as comfy as TNG, but Deep Space 9 (1993-1999) might appeal the most to modern TV audiences. It is a very bingeable show and Star Trek's most serialized series yet. It has a large, ensemble cast and takes place on a space station which is, well, stationary. As opposed to TOS and TNG, which took place on mobile starships which often encountered weird new aliens each week, DS9's crew largely interacts with the same factions in every episode. The result is a cohesive and deep experience with epic, season-long storylines.
It still has plenty of silly episodes and one-offs (it's a Star Trek show, after all), but manages to create the most immersive and complete vision of the future yet. The only known stable wormhole in the universe is parked right next to DS9, and the politicking and warring over this wormhole drives much of the action.

Benjamin Sisko

Sisko (Avery Brooks) is the station's Commander (later Captain) and accidentally becomes a religious figure to the nearby Bajoran aliens. Picard is a diplomat, but Sisko is a realist. He needs things to get done, even if his hands get dirty doing it.

Kira Nerys and Odo

Major Kira (Nana Visitor) is a Bajoran ex-freedom fighter and Sisko's second in command. Her people are recently recovering from a brutal occupation by the Cardassians. With her people free, she no longer needs to be part of the anarchist resistance. Now, Kira must adjust to newfound order in her life.
She's often paired with Odo (Rene Auberjonois), a shapeshifting alien who is the station's security officer. Odo values law and order above everything. Once thought to be the only alien of his kind, he struggles when he finds out other shapeshifters turn out to be bad guys, actually.

Miles O'Brien and Julian Bashir

TNG's transporter chief, Miles O'Brien (Colm Meaney) is back, and now he's DS9's engineer! Unfortunately, as the "relatable everyman," he's constantly being tortured and put into bad situations to make the audience feel bad. At least he's got a new best friend, chief medical officer Julian Bashir (Alexander Siddig). I feel like Dr Bashir's character took a while for the writers to figure out, but once they do he's so fun to watch. He's enthusiastic and eager and has a lot to prove the world. His parents put him through illegal genetic engineering as a child, making him super-smart, but he's still awkward as ever.

Jadzia Dax

Dax (Terry Farrell) is a Trill, an immortal space worm that lives inside a (mostly) normal guy. When the host dies, the worm can be implanted in someone else and all their memories and experiences transfer. Because of this, Jadzia is a total badass and everyone recognizes her as such.

Quark

Quark (Armin Shimerman) is a Ferengi (space capitalist alien) and operates a bar on the station. He's always trying to smuggle something and is always getting arrested by Odo. He's always looking for profit but he's usually quite pathetic, making him an endearing little guy.

Worf (TNG) joins as a permanent cast member in Season 4!

"Bar Association" - S4 E16

In this episode, the workers at Quark's bar form a union. Ferengi-centric episodes are polarizing in the fandom, but this episode is too good to miss. Space-capitalists quote Marx!

"In the Cards" - S5 E25

Jake (Sisko's son) and Nog (Quark's nephew) get into some hijinks when they try to get Sisko a rare vintage baseball card. I adore these one-off episodes that provide breaks between heavy hitting, plot heavy story arcs.

"One Little Ship" - S6 E14

This is another goofy "concept" episode-- this time, one of their shuttles is shrunk down to the size of an apple. There is plenty of action and stakes to keep you engaged, but it's also just fun to see a tiny little ship zoop around as the bad guys try to shoot them.