Well, coal can be found significantly above sea-level, so just look for an exposed vein to start with. Once you have torches, go spelunking. You will end up with more coal than you went in with, by far, if the cave is even of just average size.
Usually after finding my first coal, i just make a nice minimal small dirt cave home. Then I go spelunking. the second I get enough stone for a full set of tools, I get rid of my wooden ones and never look back. Caves make it easy to find exposed veins. Once I have quite a bit of stone, and a modest amount of Iron I go on a trip of massive deforestation. I'm talking at least 10 stacks of logs. Once I've done that, I'll have enough wood to last for quite a while, and can focus on building a nice, modest cottage, with a basement full of chests.
At that point I usually start mining. Sometimes I just start a straight up tunnel going down, other times I'll start at the lowest point of an existing cave system. The tunnel is 3x3 and that's how I start gaining ores. It's a self-fueling act. Iron for tracks (so transport time is decreased), Coal for Torches to keep lighting the tunnel. My tunnels go for 100 meters and then I build a surface access, It's usually via ladder only, though there is a junction for carts at the bottom of the shaft. However every 500 meters I have surface access via cart. Now whenever I build a surface access shaft, I build an outpost, never the same building twice.
Ideally this process would go on infintiley in all directions. However, as updates come I gen new maps, so I never get too far. And Each time I start anew I do refine the process. But that's the gist of how I play.
If I get ideas for cool buildings while mining, I will stop to go build them, however. I also spend quite some time with nothing just trying to find the perfect location to start at. Usually 2-3 in-game days.