I think it depends on the player. A lot of players sign on the first day of free agency. I have a hard time thinking they are looking for the biggest money deal.
It doesn't take long for agents and GMs to talk about money. I've read several articles in the past couple years that have painted a picture of agents telling GMs what range they're looking for, and the GM either responding with an offer or deciding they're not interested. There's not usually a lot of negotiation. You reach a deal in principle, then you work out the details. And it makes sense, given all the player movement during free agency.
And now, with the free agency warm-up period, teams and agents have a chance to talk before signings begin. So it really doesn't take long for deals to get done.
For the most part, a player is going to go for the money. If the money is equal or really close, he'll probably choose the team that's closest to contention, but some players would rather play in warm weather, or closer to their family, and so would turn down the Steelers or Ravens and go to the Dolphins or Bucs, for the same money. Some players want to start, and so turn down a team that's ready to contend to go to a worse team that's going to feature them in their gameplan (especially receivers). But mostly, it's about the money. That's why Mario Williams went to Buffalo. It's why Pierre Garcon went to Washington (our offer wasn't that far off, either). It's the general rule.
In most cases, they don't go from team to team looking for the best offer, but I'm certain the marquee free agents field offers from multiple teams, and then weigh their options with their agents. And the overriding factor, nine times out of ten, is money. And it doesn't take long for that process to play out.