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.
It might be my Dutch privilege of growing up in the country-side close to a real cheese farm, but almost every vegan cheese I've tried here has a dissapointing "margarine" taste/texture - that weird watery feeling compared to real butter. I'm sure someone will figure how to recreate something resembling the missing taste/texture eventually though. (also to be clear: it doesn't have to taste like cheese, I just want it to taste in such a way that I don't miss it, and part of that is not tasting like margarine). But again, maybe it's an issue of knowledge not being wide-spread enough to create good local alternatives.
Miyoko's Creamery also makes a fantastic cultured butter that's based on cashews and coconut oil (but with no coconut flavor). It's very creamy and hard at room temperature, much like butter and completely unlike margarine.
To do this I paid a LOT of money for the best quality vegan 'parmesan'.
It was inedible and tasted like a rubber coconut.
Yogurt is basically milk with a bacteria that eats sugar added to it. You can do that with any product similar to milk it seems. Oatly produces an oat-based yogurt, and the Swedish market has yogurts based on (at least) soy and lupine as well.
https://en.wikipedia.org/wiki/Rennet
The lactose mostly ends up in the whey, so most lactose intolerant people do okay eating rennet cheeses (and so can eat things like pizza and so on).
It's popular among the vegans I know, and some of the better pizza restaurants (in SF) even offer it for vegan pizzas.
Now I'm curious to get some Oatly and try one of Miyoko's recipes with it.