There is a board game company that is very popular & uses Kickstarter a lot, called “Cool Mini or Not”, which reminds me a bit of the kids in Stranger Things. There are a variety of excellent minister miniature dungeon crawlers (Zombicide Black Plague for feeling like super heroes vs zombies, Arcadia Quest for PvP) or strategy games (Blood Rage), or tactical combat/thematic games (The Others, which is an all vs. one game of corruption and redemption tactical combat and horror elements like Stranger Things or IMO the Buffy spinoff Angel).
There are also some other great RPGs in a box like TIME Stories, Descent, and Star Wars Imperial Assault (the latter two are nearly the same system , all vs. One style gameplay). The big excitement however is lately about Gloomhaven, which might rise to top game on BoardGameGeek next year (and deservedly).
Imperial Assault is supposedly getting an app like this soon as well.
D&D and other pen and paper RPGs have been on the rise since before Stranger Things, and I think ST making it a plot element might be due to that, more than the other way around.
We've also seen a bigger push for some 'simpler' RPG systems that are a less rules heavy to try and entice newcomers to the genre. Fate has been the big one here, but others have stepped into the ring, and Fantasy Flight Games, which is a big name in the board game world, just released the Genesys rulebooks, which are based on their 'Narrative Dice' systems, which uses much simpler rules than a more traditional d20 style system. They had previously used it for their Warhammer and Star Wars RPGs, but Genesys is meant to be general purpose for whatever setting you may like.
Same problem with trying out Pandemic: Legacy. Damn social anxiety.
Card Kingdom / Mox is very popular; if you have some anxiety, I would recommend one of the other game stores in town. Meeples Games in West Seattle is very friendly and laid back. Uncle's can be chill, too, but I have not attended an event there since the Southcenter store closed.
I recommend checking the Adventurers' League site and visiting a few events at different locations; find which one matches your interests, best. Let the staff know you're new in the area and looking for a group; they'll redirect you to an organizer who will do their best to find you a table.
Alternately, there's a plethora of local gaming conventions around Seattle. Adventurers' Leagues are usually there, too. Same drill -- show up, let them know you're new and looking for new friends.
In some ways, if the friends are new to D&D as well, it's even easier, since they don't know what's supposed to happen either!