These are great questions!
I think I would rather license the pillow out to someone else. Do I inquire with several companies at a time or do I get a response before moving on?
--- I never put all my eggs into one basket. I contact everyone and don't stop until someone makes an offer.
Can I go to a store that has it's own brand, like Target? How do I approach them?
--- Generally, the answer is "no". However, this is changing. None of the retailers are set up with any policies for licensing, so as a rule of thumb it is better to start with the distributors who understand licensing.
What do I say and to whom do I say it to?
--- I call the companies and ask the first person if they license products. and if so, what is their submission policy.
I know I don't want to sell the patent. Can I have a licensing agreement with more than one company or do I give them the opportunity to have exclusivity?
--- Generally "no". Most companies only license on an exclusive basis.
How long does something like this take?
---There is no rule of thumb here. I have done things in as little as 2 weeks, but I knew the people and had an established reputation in the industry of success. On the other hand I have known people that it took many years to license.
Is it reasonable to think I can have something in place in a month?
---That depends on who you know

I have never negotiated for anything, except on the beach in Mexico, I think this is a bit different! Please help!!!!
---That is cute! It's all about creating a win/win! This is not an easy subject to broach. If you have not already, I recommend you listen to the "Now What?" course on this site.
For more answers check out:
http://www.ideatango.com/content/section/10/32/Happy Inventing!
Lisa