I'd like to improve both my singing and speaking (mostly in everyday conversation; I'm not worried about public speaking so much) voices. There are a lot of the same issues in both areas, such as not being loud and clear enough, and the nasality in my voice that I would like to get rid of. I had to choose between improving one or the other, I would obviously choose my speaking voice, as it's the voice everybody hears all the time. So would singing lessons drastically improve my everyday speaking voice, or should I just take speech lessons?

Probably go with speech lessons. I've been singing for years and I still mumble.