Let's get this straight: you wan place that's beach-filled but mountainous, and fun and luxurious—but cheap. I'll do my best to achieve the near-impossible here. My first suggestion is to go with a Caribbean destination. There are tons of fantastic islands with mountainous terrain not too far from the beach. Second, to keep costs down you should pick somewhere that can be accessed by a lot of direct flights from the States. With those suggestions in mind, here are a couple of possibilities.
Port Antonio, Jamaica
This port town on Jamaica's northeast coast, about 90 minutes from Kingston, has never firmly established itself on the tourist map, thank goodness. As a result, its sugary beaches stay relatively empty, the resorts are cheaper, and it's not tacky or overbuilt. The Port Antonio area is also the prime base for hiking and backpacking adventures into the rugged Blue Mountains, which rise as high as 7,400 feet.
Stay: Nowhere else but Frenchman's Cove Resort, a 45-acre beachside estate (which calls itself the first-ever all-inclusive resort) where Queen Elizabeth and her royal rugrats once stayed in the 1960s but you can now get a one-room villa for $95 a night, or a three-bedroom one for $295 a night.
Cabarete, Dominican Republic
Forget the town of Puerto Plata, which has become a hive of all-inclusive resorts—where people go to enjoy the benefits of a tropical land with all of the culturally barren comforts of home. A better compromise for adventurous you and your stick-in-the-sand friends would be nearby Cabarete, slightly further east on the Dominican Republic's north coast. It's grittier than Puerto Plata, and overrun with Europeans (not that there's anything wrong with that) and kitesurfers, but incredibly lively and diverse. It's also the jumping-off point for the three-day climb up the Caribbean's tallest mountain, 11,000-foot Pico Duarte.
Stay: Velero Beach Resort is one of the Caribbean's low-cost luxury gems (although officially, it's on the Atlantic). The resort boasts an infinity pool, the suites have kitchenettes for budget-minded groups, and yes, cabana boys will serve fruity drinks with toothpick umbrellas in them at the beach, rates start at $99). Local adventure outfitter Iguana Mama will arrange your hike up Pico Duarte.