Dear Miss Snark,
Two weeks ago, I started sending out queries. I made sure that all the agents were pretty much people I would love to have represent me. So far, I've received 3 requests to read my manuscript.
My question is about timing. Do I basically just agree to go with the person who offers representation first? I can't very well say, "hold on, some other people are reading it too, let me see what they have to say," can I? (Of course, I realize that no one may offer representation, but I want to be prepared).
Actually, that's exactly what you say.
Most of the full manuscripts I read are also being read by at least one other agent. When I think it's one I'd like to take on, I let the author know. They say "it's also being read by two other agents" or "X and Y have it too".
I make sure the author knows s/he can ask questions of me, get in touch with my clients for snarkly inside info, and pitch them on the attraction of a Snark Central agreement.
If I'm the slow one off the mark, I'll get an email saying "X has offered to sign me" from an author. Usually I say Vaya con Dog pretty quickly if I know they have an offer. One less thing to read, lots of fish in the sea, etc.
The key here is time. You can't start sending partials and fulls out after you get an offer and be able to respond in any kind of timely manner to the agent who's offered representation. There was a post earlier about using an offer as leverage with other "better" agents and I took some heat for saying "no no no". You've obviously addressed that issue correctly by sending queries to agents you'd be willing to sign with.