In my experience, the best non-dairy cheeses are made from culturing nuts. Miyoko's Creamery and Punk Rawk Labs in the US both use cashews (and sometimes include other nuts) as the base, then culture them. They both taste fantastic to me, though as a 20+ year vegan I cannot compare them to dairy cheese.

I've also had recipes that used non-dairy milk as the base. None of these were cultured, so while they can taste good, they generally don't have any of the tang or sharpness of cheese. This can be fine if you're trying to make a cheese sauce but it doesn't cut it for something that's trying to be like solid cheese.

If you want to make some at home I highly recommend the cookbook Artisan Vegan Cheese by Miyoko Schinner (of Miyoko's Creamery). The recipes are actually fairly simple, though I think it takes practice to get a consistently excellent result, which is why I'm lazy and just buy the cheese from her company instead of making it myself.

You can definitely make yogurt from non-dairy milks, and AFAIK it's done with the same strain of bacteria. In the US, these are available at mainstream groceries (at least in Minneapolis where I live) as well as co-ops and Whole Foods. I've had yogurt based on on soy, almond, and coconut milk (and maybe others I'm forgetting). My favorite is probably Kite Hill's yogurt. That said, I'm not that into yogurt in general, so I rarely eat it.