If you want sushi, then it's not bad at all at Sugar Club (within Sugar Ridge), otherwise The Larder over on Friars Hill Road - just outside St Johns.
Try lunch at Keyonnas (Sugar Ridge shuttle will take you there) on Turners Beach - about 10 minutes away.
English Harbour is very lively at this time of year with the runup to Sailing Week - lots of amazing yachts in dock and lots going on in the bars and restaurants down there - for that alone it might be worth having a rental car for at least a few days.
There are a number of places to eat and drink within Jolly Harbour - just across the road - Al Porto for pizza is very good value (and they'll shuttle you there too!) Sundowners at Castaways on the beach (happy hour there) or go a couple of beaches down to Dennis at Ffryes for a fantastic view.
For a very special Sunday afternoon, have tapas at Sheer Rocks @ Cocobay - book a daybed and enjoy the pool.
For something a bit more active, head up through Fig Tree Drive, but turn off about half way up and head up to Wallings Dam - there are some great hikes starting there, or just enjoy the solitude and birdsong. http://www.antiguanice.com/v2/client.php?id=834
I'm with Mark - forget Coconut Grove, but Half Moon Bay is well worth seeing, and if you're over on that side of the island, have lunch at either Harmony Hall or Nonsuch Bay Resort - both lovely. You can get a water taxi across to Green Island for snorkelling too.
I'd thoroughly recommend taking a water based tour of the island - the Xtreme Circumnavigation would be my top choice - https://www.adventureantigua.com/the-xtreme-circumnav/
- which includes a stop at Sting Ray City, or take it easy and go sailing on a traditional Carriacou sloop - http://www.sailing-antigua.com/
Both these originate from Jolly Harbour, so right on the doorstep for you.
Congratulations, and have fun!
I'm with Mark - don't bother with Coconut Grove, but Half Moon Bay's well worth a visit