I'm not actually watching with my grandma, I just don't want over the top violence, language, or adult content. (I especially don't want the last.)
OK, here's my real suggestion: Pick up a few things in the past that you wanted to watch, watch the first couple of episodes, then decide if you want to continue watching those. If you don't want to continue those,
here are some things you can watchBuso Renkin- Its your stereotypical shounen anime, but it has some great characters and an awesome plot
Rick and Morty- Its got a great plot, awesome depth, amazing comedy, and continuity. It has some harsh language here and there, but nothing you wouldn't hear tenfold in Spam.
Overlord- Good plot, great characters, awesome setting. Has some sexual bits (first, second and fifth episodes come to mind), but they're short pieces and don't really add much. I would definitely suggest this if you HAVE to watch something
Hunter X Hunter- Awesome characters, awesome plot, awesome setting. A bit long, but totally worth it. It does get a bit graphic in the second third and fourth arcs, but its no more graphic than a t for teen video game back in the early 2000's.
JoJo's Bizarre Adventure- Great characters, average plot, less than stellar setting. OK, this one does have a bit more graphic violence than the others, but the characters and stuff are light-hearted enough to make it seem like nothing. However, if you don't want to see that much blood and stuff, stop watching after the second arc. Even if it is censored, Stardust Crusaders has so many gruesome deaths that its crazy.
Gravity Falls- Great comedy, Great characters, great plot. This is definitely a suggestion you should take. Its a really good series.
Over the Garden Wall- Awesome characters, awesome plot, awesome setting. Just all around awesome. Definitely something you should think about watching.
There's also Steven Universe, but that's more of a niche show than something you'd suggest. Plus the comedic episodes break up the plot episodes too often.