I wouldn't go around and get a bunch of Fastpasses first thing in the morning. Think of all the time you would waste!
If you get to the parks early (be on hand for opening), you should be able to ride many rides without waiting. As you exit the ride (or enter), pick up a Fastpass for later in the day. I'm not familiar with the unlimited Fastpasses--can you hold more than one Fastpass for the same ride? If so, you might want to grab 2 for your favorites so you can ride twice in row when you come back!
Another strategy you might want to use is to pick up a few Fastpasses before you head out of the park to take a break or to eat. Make sure you check the return times, and try to time it to return to the park when you windows for the rides start.
The only rides I've seen that run out of Fastpasses in the middle of the day are Splash Mountain, Indiana Jones, and, surprisingly, the Autopia, which is a notoriously slow loader. At DCA, Soarin' over California often runs out, but that's about it. The other rides will often still have Fastpasses available through the early evening, and some never run out.
Like I said, I've never used unlimited Fastpasses, but it seems like it would be easy to get out of control with them--I know I'd be in danger of that!

My biggest piece of advice would be not to run around spending all your time gathering Fastpasses, but rather picking them up along the way as needed.
Oh, one more thing, if you find you have gone wild and picked up too many Fastpasses, I'm sure another family would love to have your extras. There's no need to let the passes go to waste!
Hope this helps!
