Unless otherwise specified, the following assumes a St. John's base.
Take a walk along the waterfront to see lots of interesting ships from many countries, although the Portuguese don't come much anymore.
Visit the parks:
Bannerman Park (festival site, near downtown, near Venice pizza), Lots of green, statues, fountains, paths, a ball park, pool.
Bowring Park (on west side, a half hour walk from downtown). bigger, with green, trails, flowers, a small pond with swans, and the Waterford River. There is now a trail from there along the old railway track much of the way to downtown, at least as far as the old railway station (the true mile 0).
Pippy Park. There are many trails here, and a Rennie's Mill river restoration project, and a fluvarium. The Oxen Pond botanical gardens are not far away.
Quidi Vidi lake (regatta site). The regatta, oldest sporting event in North America, is held on the first Wednesday in August, unless delayed by weather (too much wind and/or rain). It is the only holiday decided on the morning of the event.
To quote Olaf: "Closer to St. John's, and walkable if you are in shape (since it's an uphill climb) is Signal Hill, with a Visitor Centre halfway up, and a great view of St. John's at the top, complete with interpretive displays and museum exhibits."
Signal Hill/Cabot Tower: where Marconi sent his transatlantic signal from. scenic point for watching icebergs, etc., also for lovers to park at night, so I've heard. Some trails as well. In the summer the Signal Hill Tattoo (young people in old military uniforms) do exercises. Occasional live music in midsummer. The nearby Battery has some old cannons and emplacements, and the place where the chain was anchored to cut off ships during the war.
Olaf: "If you are on foot, you are pretty well limited to St. John's." However, some bus companies, including Fleetline, run around the bay, so if you know where you are going you can get the bus to drop you off for a day of hiking/etc. I think they also do charters and possibly tours.
Cycling conditions are not that good. The roads can be bumpy, and there are few bike lanes. However, many people cycle, and there may even be bike tours along the old railway line soon.
Olaf: "The Murray Premises, near the harbour and the Delta Hotel, were once a mercantile complex which have been renovated into shops, restaurants, and" now a Newfoundland Science Centre.