I've finally been able to get all of the mushroom achieves after hours of grove hopping. It also seems to me like I found fabled flowers rather quickly, and that's partly because I knew where to look.
The purple crystal cave, autumn biome, and spiral cave are all places that you can find plentiful mushrooms.

The achievements you can get using this guide are:

"Fungus Scholar"
Found all types of mushrooms

"Mushroom Hoarder"
Found 500 common mushrooms

"Delicious Rarities"
Found 100 rare mushrooms

"Odd Picks"
Found 50 fabled mushrooms

If you're curious about mushrooms and want to know more, you can also look at Jynn's Guide , which shows the different possible mushrooms to find.

I worked out a route through the map that hit up all the places that you should go to to find mushrooms o' plenty. Using Nemo's Rough Map I drew out a nearly exact route to take.

Along the way you can also find many crystals and fabled flowers, as the areas are pretty filled with them.





With a series of grove hopping, you should be able to get a good amount of mushrooms from each grove.

You do also want to make sure that the cave underneath the mountains connecting the autumn and winter biomes is explored. That cave has been a gold mine for me when searching for fabled flowers and mushrooms.

By far the hardest of the mushroom achieves to get was this one. Unlike fabled flowers, you cannot spot them from very far away, and must get closer for them to appear on your screen.
The spawn rates for these was also seemingly pretty low.
I could find maybe 2 on average in one grove, and aroud 4~5 if I was lucky.

Here for the autumn biome (also using Nemo's Mushroom forest and foggy tree map ), I marked the spots where I had found fabled mushrooms in recurring places by red dots.





As for the purple crystal cave and spiral cave, those areas are very linear, and you shouldn't have to worry about exploring too much to find the mushrooms. All you would need to do is follow the cave from entrance to exit.

I do hope this guide was helpful. I think that I found all the mushrooms relatively quickly (I think even possibly the first to get the "Odd Picks" achievement from what I've seen)

Good luck exploring and getting those achievements!