Strategy

I live in San Francisco and can easily get to the airport by BART, so I will aim to be at SFO by about 9:30am to be eligible to catch a decent crop of the morning flights.

My tool of choice is probably going to be google.com/flights, which allows you to enter an origin city but leave the destination blank, and it gives you a world map with prices overlaid on each city.  ITA Matrix seems convenient as well, since you can search from one origin to multiple destinations simultaneously, so I could copy down the airport codes of every city on the SFO departure board and see what comes up.  This is less helpful though if I want to find cities that are 1-2 connecting flights away though, the map approach may be better for that.

I'm not sure if the SFO check-in areas have wi-fi available either, so I'll be prepared to tether through my phone to get internet (thanks T-Mobile for being the least dick-ish carrier in that respect and not charging me for this).

As for airlines, I'm going to initially limit my search to Star Alliance partners due to having status.  I realize this is quite an arbitrary restriction, which I may lift, but if the destinations work out, this seems beneficial.  In particular, this seems like the kind of situation in which you may find yourself in a bind or stranded, and having the airlines pay more attention to you due to status seems like a good thing to have in your back pocket, especially since this is my first time doing this.  Also I'm anticipating particularly long layovers and having lounge access should help me avoid becoming crankier than usual.