If you do end up with a car, I'd try By the Beach for fish 'n' chips rather than Ches', which has coasted on reputation for decades.

The view from the Rooms cafe is great even if you don't have time for the whole museum/gallery. The cafe has a partridgeberry cheesecake which is nice.

Decent beer/atmosphere at the Yellow Belly; OK beer but better casual vibe at the Ship or the Duke.

If you drive down the southern shore (to go whale watching), you might want to continue on to Ferryland. Interesting archeological site, and very pleasant picnic lunch option at the lighthouse:
http://www.lighthousepicnics.ca/

I second other comments: walk down from Signal Hill the long/harbour way, or check out Quidi Vidi, go on a whale/puffin/iceberg boat trip, check icebergfinder.com.

And pack warm clothes. June can be chilly!