The Galapagos DOES NOT have what you are seeking. It's one big "National Park", for the most part yo

Basically, we want to be able to relax on a beach that's not touristy, and enjoy a few excursions like hiking and fishing. We also are looking to do this on a budget (hence the "nothing fancy" ford rent a car for our accommodations). My fiance has never been out of the country so we don't really want to do Hawaii. I've considered Belize, Grenada, ford rent a car Tobego, Guyana, the Galapagos, St. Lucia. ford rent a car All seem great but I'm worried about the weather in June/July.
June or july is going to be warm anywhere in the Caribbean but it will also be more humid when compared to the winter months. I wouldn't worry let the weather drive your decision - it will be essentially the same no mater where in the Caribbean you go. - expect day time temps in the mid-to high 80's with lots of sun and an occasional brief passing tropical shower (most last only minutes).
The Galapagos DOES NOT have what you are seeking. It's one big "National Park", for the most part you have to go on guided group tours to explore the different islands so you won't find the seclusion you are looking for nor will you find beautiful beaches. Can't comment on Tobago or Guyana as I've never been to those places.
Of the places you mentioned Belize, Grenada and St. Lucia will have what you want. You might also look at Roatan (it's still somewhat undiscovered so you'll find the seclusion you want at a price that may fit your yet undisclosed budget.
Have you considered a cruise? They are reasonable ford rent a car when you figure your accommodations meals, ford rent a car and entertainment are all included. You can get off at the Islands and wander the beaches, and also do sightseeing. You can leave right from Florida. Just an idea!