The Next Generation

If you're new to the franchise, The Next Generation (1987-1994) is where I'd recommend starting. The first few seasons are a little rough (something that would become a trend for future Trek series), but this show is the ultimate comfort watch. With a few meta-plot exceptions and some two-parter episodes, the series is episodic, meaning you can safely jump around episodes. This also means every episode is neatly resolved by the time credits roll. If the universe is about to end at minute 39, you can rest assured it will be saved by minute 44. Everything will always be okay, every time.
Being the last series that creator Gene Roddenberry worked on, it has some charming holdovers from its '60s past. Chief among this is the edict that there should be no conflicts between the main characters. That means that Star Trek's utopian vision of the future extends to the workplace: could you imagine if all of your coworkers got along and supported each other? Crazy!

Jean-Luc Picard

Captain Picard (Patrick Stewart) is starkly different to Captain Kirk. Where Kirk is the center of action and attention, Picard often delegates tasks to the most qualified person. He's the best boss one could hope for: supportive, respectful, and willing to listen. Reserved and even at times cold and distant, Picard is an expert diplomat. It might be more fun to serve on Kirk's ship, but, for my money, Picard will keep me alive.

Data and Geordie La Forge

Geordie La Forge (LeVar Burton) is the ship's chief engineer and Data (Brent Spiner) is his android best friend. Every scene these two share is a comfort. Data, an android yearning to be human, struggles to grasp social situations (he's coded as autistic) and he commiserates with equally awkward Geordie.

William Riker and Deanna Troi

Riker (Jonathan Frakes) is Picard's second-in-command and provides a swaggering action-hero panache to the bridge. Troi (Marina Sirtis) is a telepathic ship's counselor and Riker's old flame. There's a bit of will-they-won't, but they know how to keep things friendly and adult. They are professionals after all!

Beverly and Wesley Crusher

Beverly Crusher (Gates McFadden) is the ship's chief medical officer and mother to Wesley (Wil Wheaton), a child prodigy. Dr Crusher is such a nurturing person, always encouraging others and generally being the ship's mom. Wesley is a bit of a controversial character in the fandom, but I love him. He is an audience stand-in-- a kid bewildered by the fantastic sci-fi world around him. He's a nerdy teenage genius who fulfills our fantasies.

Worf

Worf is a Klingon, the warmongering samurai aliens introduced in TOS, and a fan favorite. He was raised by humans and is essentially a weeaboo for Klingons. He's read their codes of honor and mythologies and tries to live his life as honorably and traditionally Klingon as possible, all while paradoxically serving aboard a human ship and acting nothing like most Klingons. Worf later appears in DS9 as a series regular.

"Data's Day" - S4 E11

This is the one. The comfiest hour of TV ever created. We get to follow around Data as he goes about a typical day aboard the Enterprise. No big space battles or epic stakes. Just hanging out with your TV friends for a bit.

"Disaster" - S5 E5

The ship gets a flat tire and the characters are stuck separated from each other, forcing everyone to work outside of their comfort zones. Which, in Worf's case, means delivering a baby.

"Cause and Effect" - S5 E18

The crew is caught in a time-loop and the ship is destroyed every commercial break! This, along with episodes like Parallels (S7 E11) where characters get stuck in loops are, for some reason, such a comfort to watch.