My nieces, who are between the ages of your kids, loved The Good Place, Parks and Rec, and Brooklyn 99. They are all warm-hearted, non-violent, and low on salty language due to being network shows, but some of the humor involves sex so YMMV.
Better Off Ted (Hulu) is a delightful, slightly older comedy about a guy working for a totally absurd corporation.
Agent Carter (Disney+) is a great adventure series centered on a strong female character (and has excellent costumes)
BrainDead (CBS All Access) is a comedy/drama series about mind-controlling aliens invading Washington DC. There is some splatter gore but it's really played for laughs...you would probably want to check this one out first but my nieces would be fine with it. iZombie is similar--Rose McIver is amazing as the zombie detective who solves cases by eating victims' brains (and temporarily taking on parts of their personalities), written with a lot of humor but there is some mild gore involved.
Bunheads is a great series starring the amazing Sutton Foster as a Las Vegas showgirl who moves to a small town to teach dance lessons after being very briefly married and then widowed. For that matter, Gilmore Girls, from the same showrunner, could also be a fun teen/adult watch.
Eureka and Warehouse 13 were science fiction shows with a lot of humor--very family friendly--from a few years ago. They tend to come and go from streaming but I think are both available at the moment. Fringe was darker (likely OK for 14, might want to view first for 10) but aired on broadcast TV so it was not too extreme. It was very well-done science fiction. Timeless was a good time-travel show that aired fairly recently that was not too heavy.
The remake of One Day at a Time to feature a Cuban-American family (Netflix, and now moved to another network) is just delightful, and Rita Moreno still has more rhythm in her little finger than several current Broadway stars combined. They do explore issues relating to the teen characters, which means some discussion of sexuality, but it is extremely well-done.
I don't watch a lot of reality TV, but I know many families enjoy series like The Floor is Lava, Great British Baking Show, Master Chef Junior (which is much more pleasant than the adult version), the Amazing Race, Dirty Jobs, etc etc.