labrador_ice has given some great suggestions. I would add:
There is lots of great hiking on the Bonavista peninsula. Hike Discovery is a non-profit group that coordinates trail development and maintenance. You can find trail maps to purchase and pictures on their website. The Skerwink Trail is probably the best known and is very popular during the summer. The folks at Fishers Loft will point you in the right direction, it runs right past the property. It can be jammed at times, the nearby Fox Island trail is equally as scenic and much less busy. I think all of the trail starting points show up on Google Maps and you'll find free hiking maps locally. While you're at Fishers Loft, you can walk to the Port Rexton Brewery and sample some of their very good beer. They don't have food themselves, but there's a great food truck there that does amazing grilled cheese sandwiches. While you're driving around the peninsula you might want to stop in at the Bonavista Social Club Fabulous little place with a wood-fired pizza oven in a tiny village of about 40 people. I highly recommend it.

The Black Sheep Pub is another great place for live music in St. John's. It's small, but they have a wide variety of live music every night. You might see a great local singer/songwriter, a traditional band, or an open mic night. Florian Hoefner, an amazing jazz pianist relocated from NYC plays there occasionally.

St. John's punches well above its weight as far as restaurants go. Two "don't miss" ones not on labrador_ice's list are The Adelaide Oyster House in downtown and Mallard Cottage in Quidi Vidi. My favourite walk is through the Battery area of St. Johns to hike the Signal Hill trail to the Cabot Tower. From there, hike along the coast to Quidi Vidi stopping for a beer at the Quidi Vidi brewery or Mallard Cottage for dinner (or their beer garden.) It's about an hour to an hour and a half, depending on where in town you're starting. About an hour's walk back along Quidi Vidi lake or a quick cab ride. No Ubers in St. John's, but lots of cabs. The Merchant Tavern is owned by the same people as Raymonds - less formal, less expensive, but also very good. There's a very well-travelled local woman who writes The Food Girl in Town, a great blog that's got a lot of good reviews of St. John's restaurants and other cool information.

And to answer your questions about the Rooms, for a visitor it is primarily an art gallery/museum. Well worth seeing, try to have lunch in their cafe. Very good food, but a spectacular view out the narrows.

Many things in Newfoundland are season-dependent - whales, icebergs, festivals, park closures and operating hours, etc. When are you going?