What is the best way to approach organizing the books, so I don't have to put them on the shelves more than once?
When I moved, I used the "piles on the floor" approach. I sorted my books into large piles based on the way I tend to think about my books. For me, those categories were: Fiction--all sorts; reference--encyclopedias, dictionaries, almanacs, language guides, travel guides, atlases, etc.; poetry and plays; manga, comic book anthologies, graphic novels; craft books--knitting, weaving, etc.; history; cookbooks; assorted non-fiction; and so on.
I then shelved one category at a time, deciding if there needed to be additional sorting within a category. With the exception of fiction (I shelved by author, and to hell with genre), the comics (volumes in a series are kept together), and history/assorted non-fiction (anything related to science or history thereof got separated out), I didn't worry too much about further refining the sort.
The trick is to organize to the level at which you mentally categorize your books. My concept of "reference" is pretty broad, and I lump poetry and plays together, since I tend to think of them as "not-non-fiction-but-not-prose." My dad, on the other hand, is a theater junkie, and would probably categorize plays by period and author. My mom would probably have seventeen different criteria for categorizing her cookbooks.
Books I know I'll take out frequently (cookbooks, reference, re-readable novels, etc.) are placed where I could get to them most easily without having to hunker down or stand on tip-toe.
Also, since my shelf space is at a premium (and because I know darn well that I'll be acquiring more books), I stacked most books at the backs of the shelves, then propped another book in front of the pile that would give me an idea of what was behind there. That way, I was able to squeeze in more books per shelf-foot, and I was able to display the covers of particularly pretty books. Does that make sense?