No don't dig in there to separate it - the stalks are coming up from a potato-like tuber. You can't separate them at this point. In the fall when they die back after your first hard frost, you can lift the whole mass of tubers and clean/divide/store them. There are detailed instructions and pictures at dahlia.com or in the archives of this forum.

You can increase the size of blooms by disbudding or disbranching - removing some of the buds or branches to put more energy toward the remaining buds, but that is something more usually done by those who want to take their flowers to a competition. I'd say keep it watered, and make sure you are using a fertilizer low in nitrogen, like a tomato fertilizer, bone meal, or I use Bulbtone. There are lots of opinions on this also, some people use Miracle-Gro for example, but most dahlia websites recommend using a fertilizer that is lower in N than P and K. Good for you, I hope you got a super deal on the price too!