Well shucks, I can't believe nobody chimed in here.
So what you're looking for is a non-binary bitmap or tilemap. Godot has only the binary version built in which means you either have tile, or no tile. Does not play well with multiple types of tile. You probably won't find much on it because the solutions are varied, depends heavily on your art, they're not intuitive, and they require a lot of tinkering to get how you want it to look. Fixing problems with them can often either be done by the artist, or by the code. It's really crappy to make a tutorial for. So, just to make sure we're on the same page. I am currently using 2x2 autotile bitmasks, my artwork purposely extends half the width of my tiles, I am not using angles, it's all squares. Basically, I'm making a top down map with multiple biomes using a a tileset like this
The first thing you have to accept is that if you're going to do this you need, at minimum, four layers of tilemap. Because at any point, 4 different biomes "or blocks, or whatever" are going to meet in a corner and if you want things to look nice and "spill" into the tiles next to them they have to overlap.
The way I have been tackling this is basically imagine your grid that you already have, and make a second "artwork" grid (4 layers deep), and shift it, down and to the right, half a tile. This makes it so that every tile in your main "grid" overlaps 4 "artwork" tiles. When you change a tile in your main "grid" You're going to change the corner of each "artwork" tile you're overlapping to make one complete tile. If you start filling out your artwork tiles and you find something already there, just move to a higher layer of artwork node.
I highly, highly recommend getting real warm and fuzzy with tilemaps. Specifically how autotiling actually works. This link https://michagamedev.wordpress.com/2018/02/24/181/ is a fantastic dive on how the logic of autotiling actually operates.