Is there such a thing as a totally customisable theme?

Truly I am the ultimate virgin with WordPress. I have installed Wamp and WordPress but the default theme, Twenty Thirteen leaves much to be desired when it comes to the visualization I wish to show (for the lack of a better word).

I guess what I am looking for is the ability to build a site from “scratch”. With a single color/textured full page background, overlaid image and desired “links” to the content that I want my visitors, customers and clients to see. I really don’t think I need a full site “daily blog”, detailing my actions and whatnot.

Would all you experts here PLEASE point me in the direction I am looking to go? I do have some rough ideas saved as JPEGs That I could share if someone is willing to point me.

It does need to be easily edited and published rapidly though. This is the reason I chose WordPress for this endeavor.