Using Paint Tool Sai I work on the logo first. I use thick lines to make sure this will stand out even when I add the background.
I add the background using thinner lines to make sure the logo stays as the most prominent part of the design. The background is also done in a separate layer to make sure they stay safe even if I mess up the logo.
Adding colors. I want to make it look like one of those 70's doodles. I add colors to the logo first. 
I add colors to the backgound. Looks good to me but there's somethat's not right. The colors of the background looks too colorful and bright. Looks fine but the logo doesn't stand out in my opinion.
 
Since the background artwork is on another layer editing it is easy, I adjusted the opacity a little bit to make the logo stand out.