Costa Rica is great for a honeymoon, and April is a good time to visit as the weather is warm and sunny. For the stopover in New York, you can fly with Continental Airlines to San José via Newark for approximately £800 per person including taxes.
The budget is reasonably tight, but the easiest way to reign in your costs would be to limit yourself to just a couple of destinations within Costa Rica.
In terms of the itinerary, Arenal is definitely a must, the views are spectacular and it is a great place to go walking. You could then consider the Tortuguero National Park on the Caribbean coast, an area of mangrove swamps brimming with wildlife, or the Monteverde reserve is only a short distance from Arenal and has an incredible ecosystem of over 320 bird species, 2,000 plants including wild orchids. Mammals include monkeys, sloths and jaguars. If you would like to include some time by the beach you could consider the Nicoya Peninsula: the hotel Punta Islita would be perfect for a bit of luxury at the end of the trip. The beauty of this lodge is that you can relax on its secluded beach or by the pool or if your are feeling more energetic, there are a vast array of activities in the surrounding rainforest including trekking and horse riding.
A 10-night holiday visiting the Arenal volcano and Punta Islita and including international flights, transfers, accommodation and excursions with an English-speaking guide costs from £2,035 per person.
See our Private Journeys to Costa Rica for more ideas