macmillan,
First of all, congratulations!
Regarding which town to stay in, I definitely agree with Roberto that Monterosso would be the best choice. It's the largest of the five towns and therefore has the greatest number of accommodation choices, as well as good restaurants and other amenities. It also has the nicest beaches of the five towns.
Yes, you guessed it, August is peak travel time there. I would NOT suggest "winging it" and it may already be too late to find accommodations there. I often start booking in February or March for travel in September, and even that early it can be a challenge finding something. If you don't have a copy of the Italy Guidebook available, you can start with the accommodations on THIS list which is reasonably current AFAIK. You may have to adjust your budget and take whatever you can find.
The two best places in town (Hotel Villa Steno and Hotel Pasquale) are likely booked solid, but you can also check Villa Adriana or Hotel Porto Roca (which is a bit on the posh side but incredible views from their patio). If you can't find anything in Montrerosso, you could also try Levanto, as it's only about four minutes by train from Monterosso and also has a great beach.
If you DO manage to find something and need further information on the area, post another note and I'm sure the group here will have lots of tips to offer.
Buon Viaggio!