My wife and I are veteran Latin American travelers. She is Chilean and our 3 daughters are all dual-citizens so we normally travel to Chile on vacation but last June we decided instead to travel to Costa Rica with my parents. Over the years either or both of us have lived and worked in Guatemala, Argentina, and Brazil in addition to of course Chile where my wife grew up. So our level of comfort with independent travel in Latin America might be higher than most. In any event, after a LOT of research this is the 2-week trip that we did in Costa Rica.
Week 1: Fly direct from DFW to Liberia Costa Rica. Liberia is a smaller city in the NW of the country but with a large modern airport that caters to US and Euro travelers. It's a good alternative to San Jose. From Liberia we rented a 4x4 car from a local car rental company Vamos: http://vamos4x4.com/vamos/
which provided good service and a reliable Mitsubisi diesel 4x4 that was a couple years old but in good shape. I brought my Garmin GPS along and had bought and downloaded the Costa Rica maps from EZFind which is the best Garmin highway maps for Costa Rica: http://www.navsatcr.com/ezfind.php
I would strongly advice not even contemplating driving in Costa Rica without new turn-by-turn maps of the country. Highway signs are sporadic and often covered by trees. I would have gotten lost many times trying to drive with a paper map on my lap through rural Costa Rica. The Garmin was flawless. From Liberia we drove to the Pacific Coast town of Puerto Carrillo where we spent a week at this VBRO house owned by a nice couple in Colorado. http://www.vrbo.com/220000
The house was absolutely fabulous and the caretaker is connected to everyone in town and so can arrange any excursion you could want from surfing lessons to kayaking. We did surf lessons, kayaking, visited a sea turtle reserve at night to see them laying eggs, and did lots of hiking around and quality beach time. Monkeys visited our house every day.
Week 2: We drove from Puerto Carrillo to Arenal where we spent 7 days at Leaves and Lizards Resort up high in the mountains overlooking Arenal Volcano. http://leavesandlizards.com/index.htm
This is something of an eco-resort run by an American couple from North Carolina (I think). The setting is fabulous. You rent a cabin on their property and meals are provided group style in a main restaurant. They will arrange all manner of excursions but you really need your own 4x4 to get around. We did zip lining, a day-long horse back riding expedition into the forest to a private waterfall and pool (highlight of the trip for the kids), a day on a Costa Rican farm, a day at a rescue zoo, a day-long trip to Rio Negro for a river tour to see birds, monkeys, crocs etc (highlight of the trip for my wife and me) and some poking around Arenal (hanging bridges, shopping etc.). A week might have been slightly long here for me but the kids absolutely loved it.
My general recommendation. With kids I think it is better to pick one or two locations from which to make a base and not try to do the whirlwind tour. Kids get bored driving around. For a 1-week trip just pick one location and for a 2-week trip just pick 2 locations. The country is small enough that you can pretty much do day excursions to anywhere and still get back to your home base at night so there really isn't a lot of reason to keep packing up and changing lodging. I would pick one beach location for one week (probably on the Pacific). June is the rainy season so the NW part of the country in Guanacaste is probably the best bet as it is the least rainy. We went in June and weren't bothered in the least by the rain. It came down for a short time on a few afternoons but didn't affect us. Then for another week pick one mountain location. The Arenal area has the most to offer in terms of mountain and rainforest tourism. People also like Monteverde. Pick one or the other, you don't need to do both. Arenal is probably better for kids as there are more kid friendly options. Monteverde is probably better for hard-core birding and naturalist types. But they are similar.
I will pm you with a link to our photo albums if you are curious. Basically there is nothing I would have done differently. It was a splendid trip.