In St Johns I would suggest a visit to The Rooms.
http://www.therooms.ca/
A great introduction to the history of Newfoundland. The restaurant there has the very best view of the harbour.

Of course you should go to Cape Spear. You can do a nice hike from here on the East Coast Trail
http://eastcoasttrail.ca/

The root cellars and potentially puffins at Elliston are interesting
http://www.rootcellars.com/
and will give you some time to stretch your legs and get out of the car. Lots of wandering!
We loved Bonavista too.

You would not be able to stay in the same hotel for the stay or would have to backtrack every night.

Thanks for the reply. I'll make sure to put those on my list of must see.

Is St. Pierre a crucial part of this trip? It'll be about a 5-hour drive from Bonavista to Fortune, then an hour on the ferry, on Day 4. Then the next day you'll take the ferry again and drive at least 3 hours to Argentia. (btw, why ARgentia?) St. Pierre is an interesting anomoly, but you will only have an chance to whip around the town.
I would make a two-point stay: St. John's (with the Rooms and Cape Spear, as irecommend says), Signal Hill, maybe the Fluvarium if your daughter would like being under a river,and a downtown walk. For alternatives, you could look at Newman's wine vaults, the Basilica, or Quidi Vidi. Then I'd do day trips to Bay Bulls on the Southern Shore for a whale/puffin (I don't know what month you're planning on) boat trip, and include the archaeology dig in Ferryland. Another day could include the remains of the French fort at Castle Hill in Placentia, and the birds at Cape St. Mary's.
Then after 4-5 nights I'd leave the Avalon for the Bonavista Peninsula. Bonavista has a couple of small museums, a lighthouse with a great view, the Dungeons Park (sea caves). Nearby Trinity has a lot of restored historic houses and churches, plus the outdoor Trinity Pageant is a "walkthrough" play about Newfoundland history that has enough movement to appeal to a 5--year-old.

I agree with Bay Bulls as an excellent spot for whale and puffin watching, if you are there at the right time of the year. Ferryland is also most interesting, and not too large a site so manageable for a five year old. There is a lighthouse high on a cliff nearby where you can have lunch served picnic style as you sit on the grass overlooking the bay. Quite charming. In general, I think you will find Newfoundland and the locals sufficiently entertaining that you won't have trouble enjoying your visit,and no need to pack your agenda.

When do you plan on visiting NL? That makes a big difference in what I might suggest you do.