Unless otherwise specified, the following assumes a St. John's base.
Olaf notes: "If you have or rent a car, then some of the nearby outports (Petty Harbour, Goulds, Bay Bulls, Ferryland to the south." For a day trip you could continue on out to Trepassey, St. Mary's Bay, Placentia and Argentia, and come back on the highway. Or it might be better to drive out on the highway and then come back along the Southern Avalon, since it is usually cooler and foggier there, and by the afternoon it would be warmer. In season, you will be able to spot whales from shore. If you get the Sydney--Argentia ferry either way, this might be a nice leg. And note that The Irish Loop contains enough attractions that it could be not just a day trip from St. John's but you could stay in the area for a night or two. One of the main attractions is the Colony of Avalon archaeological digs in Ferryland.
Olaf notes: "To the north, Quidi Vidi, Torbay, Pouch Cove -- pronounced "Pooch" are worth a visit; look for Marine Drive to the north of St. John's, especially Red Cliff Road for a spectacular view of Logy Bay, though be warned that the 2-3 km road can be quite rough in places. But the view is worth it. The Flake House Restaurant in Quidi Vidi is excellent, though pricy, and reservations are recommended."
Also check out the sedimentary geology at Flatrock beach, and the nearby shrine visited by the pope a few years ago. Portugal Cove, St. Thomas and St. Philip's are also nearby.
And of course, you should do a day trip around Conception Bay. If you drive out on the coastal road, you will pass through Paradise, Mount Pearl, Topsail, Manuel's, Foxtrap, Kelligrews, Seal Cove, Holyrood, Lakeview, Chapel's Cove, Harbour Main, Avondale, Colliers, Conception Harbour, Marysvale, Brigus, Bay Roberts, Carbonear, Harbour Grace.
Manuel's has an amazing Trilobite shale deposit, easily accessible, so go get some fossils. There is also a nearby trail to the beach.
Through CBS (Conception Bay South) there are some good views of the water and the island.
Holyrood has a little local park and swimming hole, on the highway access road. There is also a salmon fishing river, but there might not be many fish these days.
In Chapel's Cove there is a beach suitable for catching caplin or collecting rocks. There is also a nearby cove called Red Rock Cove which is more secluded, but which requires a short hike and climb. There is a nearby bed-and-breakfast (run by some friends of mine), Kennedy's Country Corner, (709) 229-6568, see the Accomodations section for more detail.
Harbour Main has a beautiful little church with a hand-carved altar inside. I'm not sure if this is accessible to visitors at any time other than regular services. The former school was recently converted to a community centre, and there is some discussion going on about making the former priest's house into a museum. There is also a local ocean-river juncture (inside a small breakwater) swimming hole over near Albert's Lounge.
Avondale and beyond: There is a local falls, with a swimming hole. Beyond Avondale the road becomes more winding, narrow and bumpy, but there are lots of good view points and small parks. There are some very old houses in Brigus, Carbonear and Harbour Grace, especially on the side roads near the water. Some pirates and quite a few fishing captains built mansions here.
Cape Spear is also nearby by car (about 15 km); it is, to quote Olaf, "the easternmost point of land in what is technically North America." Some hardy folks go there at sunrise on the solstice(s), or perhaps the first sunrise after the winter solstice, to greet the returning sun with a thermos and blanket. During late summer there are also some wonderful folk music concerts there put on by Amber Music.
Another good trip is to take the ferry over to Bell Island and check out the old mineworks, and the view. Also First Light Productions runs a theatrical historical show and tour though I think only during tourist season.
Butterpot Provincial Park, 15 miles away, allows camping, has trails
Gushue's Provincial Park (will be privatized soon), 40 miles away, also allows camping
There are probably closer official or unofficial spots for camping, including Pippy Park. My brother sometimes went camping on the South Side Hills, but someone stole some of his stuff once when he left the tent for a long period.