I trained the gun on the clearing and braced myself, muscles tensed, nerves jangling, finger on the trigger. Suddenly a mongoose burst out of the bushes and made a break for it. I fired – bang! – and it exploded into a dozen pieces. Success.

Of course, it wasn't a real mongoose. It was a clay disc, designed to race across the ground just like the mongooses that dash across Antigua's perilous roads (and often meet an equally sticky end). I was trying my hand at clay-pigeon shooting at the Reservoir Range activity centre in the south-east of the island; the centre is one of the very few places in the Caribbean to offer the sport.

English country pursuits may seem a little incongruous in a West Indian setting. Antigua is better known as a luxury beach-break destination: it claims to have a beach for every day of the year and attracts a jet-setting yacht-going crowd.

The beaches are certainly beautiful – such as Half Moon Bay, a wide sweep of white-gold sand backed by lush vegetation, with a single beach bar, Smiling Harry's, that serves simple chicken-and-rice dishes and frosty bottles of Wadadli, the local beer.

But what if you get restless after a few days of lying on the beach? Antigua caters surprisingly well for active holidaymakers. Also at Reservoir Range, my boyfriend John and I had a go at archery – much harder than it looks – and mountain biking. We chose an off-road trail around the Potworks Dam reservoir, passing farmland and a donkey sanctuary, and revelling in the chance to explore the interior of the island, away from the resorts. Reservoir Range also plans to add more activities, including fishing, rowing and other kinds of shooting.

Elsewhere on the island, water sports are big business. Kite-surfing hotspots are Jabberwock Beach in the north and Nonsuch Bay in the south-east. We tried it out at the 40 Knots surf school () at Nonsuch Bay, which co-hosted the island's first kite-surfing festival earlier this year. We sped out to tiny, uninhabited Green Island – "the office", joked our instructor, Irene – on a stomach-churningly bumpy motorised dinghy. We practised controlling the kite on land and at sea – easier said than done with a three-metre-wide inflatable kite in a brisk wind. At one point, John lost control and was yanked off the boat, landing face-down in the sea and getting a lungful of salt water.

The follow-up lesson focused on body-dragging: using the power of the kite to move through the sea. We were soon tearing about the water at top speed. Then came the "surfing" part, using a board similar to a snowboard. By the end of the lesson we had both made it upright several times – very briefly, admittedly, and always preceding an epic wipeout – and felt quite proud of ourselves. On my final attempt, I stayed up and in control for a few long, glorious seconds, but the buzz stayed with me for hours.

We also tried stand-up paddle boarding (SUP). You start by kneeling on a surfboard, propelling yourself along with a single-blade paddle. It is surprisingly hard work, especially on the thighs. When we'd got the hang of that, we stood up shakily. Turning is the hardest part. I heard a succession of splashes as John repeatedly fell in, but the smug smile was wiped off my face when I found myself swimming, too. SUP isn't as exhilarating as kite surfing, but it is a lot of fun and gives your limbs a good workout. And the flat, shallow water of Nonsuch Bay is the perfect place to learn.

You can't visit Antigua without going sailing: we opted for a romantic sunset sail around the bay. The island is saturated with sailing culture, from the Maltese Falcon (one of the biggest sailing yachts in the world) at anchor in Falmouth harbour and the regular regattas to the yacht crews filling the bars. Then there's the island's seafaring heritage: the biggest tourist attraction is Nelson's Dockyard in English Harbour, where the admiral himself lived in the 1780s.

On a rare cool day, we climbed to Antigua's highest point. Mount Obama (402m), the biggest of the Shekerley mountains in the island's south-west, was renamed in honour of the US president in 2009 – it undoubtedly has more of a ring than its former name, Boggy Peak. Escaped slaves gathered there in the late 17th century, hiding out on the forested slopes before setting up a maroon camp.

The government is establishing a national park in recognition of the site's cultural significance, which will hopefully enhance the visitor experience. At the moment, the path up the mountain is difficult to find and the views from the top are restricted by a fenced-off telecoms compound. Still, it was a decent short hike – extremely steep in places – with a good view to justify the effort.

Other options for active visitors include snorkelling – try Deep Bay, where there is a 100-year-old wreck; diving at Cades Reef (jollydive.com); exploring old fortifications such as Fort Berkeley; and taking the ferry to neighbouring Barbuda to see the frigate bird colony, caves and pink-sand beaches. Thrill-seekers can also try a zipline tour in the rainforest (antiguarainforest.com).

Much of the accommodation on the island is expensive and all-inclusive, which discourages people from leaving their resort. Nonsuch Bay, where we stayed, is unusual in offering luxurious but reasonably priced (by Caribbean standards) self-catering accommodation at Nonsuch Bay Resort. Our apartment had a four-poster bed, Jacuzzi bath and wraparound veranda with sea views. There is also an excellent restaurant on site – the chef, Mitchell Husbands, is a past winner of the Caribbean Chef of the Year competition. We enjoyed delicious West Indian breakfasts of okra "chop-up", saltfish, spicy sausage and johnny cakes. Dinner could be jerk pork chop with black beans, followed by local black pineapple, grilled and served with rum and raisin ice-cream.

By the end of two tiring, activity-filled weeks, a relaxing beach break was actually starting to appeal. Maybe after another go at that mongoose …

• Accommodation was provided by Nonsuch Bay Resort (+1 268 562 8000, nonsuchbayresort.com) which has apartments from £126 a night for two. BA (britishairways.com) flies from Gatwick to Antigua from around £550 return