Hey Erin. First let me thank Jennifer and Nichole for the mentions. It does make me feel good knowing I've helped people with their travels, especially the first-time travelers for which this can be an overwhelming experience.
I am on file with Fastaff and their subsidiary (U.S. Nursing) and do echo the sentiment of Lindsay. However, my recruiter at Fastaff does work with first-time travelers and although they do a lot of assignments offering 48 and 60 hr work weeks (NOT something you should take as a first-time traveler), they do offer the standard 13 week (36 or 40hr) contracts. It's really all about being comfortable with your recruiter and trusting your gut. Have you asked your recruiter at Fastaff the questions you posted here? That might help provide answers, but also help you gauge your relationship with your recruiter. If they spend the time to work with you and answer your questions...trust builder. If they reassure you that "everything will be fine"....RUN!
First, there is NO company out there that is the best, only the company that is best for you (ok, I apologize, that was kinda corny to say, but also true). You do have to figure out what you need.....good benefits, higher hourly rate, better housing? Don't pick a company based on highest hourly rate. You need a well-rounded company that will also back you when there are any issues while on the road. This is especially important since you will be traveling with your son.
Childcare can be complicated, but whenever you are offered a position, quickly see if you can find places you would trust to take care of your child. Kinder Care and other chains are nationwide, but you might need to limit your assignment selection to urban areas or suburbs to give you more childcare options.
For your first assignment, take the housing they provide! You do not want the added burdon of trying to arrange housing added to all the other stressors of a first-time assignment. After you have been on the road a few times, you can explore this option more, but don't do it on your first assignment.
Most all companies provide a free one bedroom apartment, furnished. If you need a two bedroom, most companies can accommodate you, but you will have to pay extra each month (this rate will vary, so ask your recruiter what this would run). Stay away from extended stay options as this is not great if you're traveling alone and would certainly not be ideal with a child.
As for your schedule, ask this in the interview with the facility. Most hospitals will allow travelers to self schedule along with the regular staff.
Let your recruiter know that day shift is your only option and that will save their time and yours.
Finally, join more companies! When searching for jobs, nobody goes to one company and then waits to hear from them for a job, they submit resumes all around town; travel nursing should be no different. I know that it is a pain to do all that paperwork they require in the beginning, but it will open you up to more assignment offerings.
I work with a lot of first-time travelers and if you would like the names of the recruiters I use at the companies I trust, feel free to email me at david@travelnursesbible.com. As Wes indicated, it's nice when travelers will drop my name to get me a referral, but it's also pretty damn cool to get those emails from travelers I've helped secure a first assignment.
I hope this helps answer some of your questions and again, feel free to email me at david@travelnursesbible.com if you need more help.
David
http://travelnursesbible.com 
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