Caribbean for a Week Before Mexico: Stick to One Island or Island-Hop?

Me and my friend (both 22F) have a week to spend in the Caribbean this October before we head to Mexico. We’re on a backpacker’s budget and can’t decide: should we explore just one island thoroughly or try hopping a few? We’re looking for:

• Budget-friendly options: Cheap stays, eats, and fun activities.
• Ease of travel: Especially since we’re heading to Mexico after. Is there an island that makes this easier?
• Activities: We’re up for anything fun and unique to the Caribbean experience.

Questions:

1. For a week’s visit, is it better to stick to one island or do some island-hopping? Why?
2. If you suggest one island, which one? If you suggest hopping, which islands and in what order?
3. Any specific tips on transportation, places to stay, or must-do activities on a tight budget?

Looking forward to your suggestions. Thanks!