If you're talking just 3-4 days, you won't be able to see too much of the province. The island is really quite large.

I'd say a couple of days in St. John's are really a must. Do a hike on Signal Hill, go out to some of the bars on George St. or Water St. to check out some music and talk to the locals, perhaps check out Quidi Vidi and the brewery. It's a nice town, and people are very friendly - although be warned, if you're not into heavy drinking, you might find the nightlife a bit intense.

You can also visit some scenic outports (small fishing towns) a relatively close drive away from St. John's. Trinity is the most famous one, but you could also hit up some other places as well if you rented a car.

The Burin is also nice - I spent a few days a couple of years ago in a beautiful B&B in Grand Bank - but it's a long drive from St. John's and probably not worth it unless you've got 5-7 days.