There is so much to see in Costa Rica with kids, way too much to cover in a single trip. I suggest you choose a couple of activities and plan your vacation around them.
It is easy to stay in San José and take day tours from there. There is plenty to see within easy driving distance, but the city itself is not that exciting. Some of our favorite places have been small, less touristy towns far from San José. These destinations require several hours of travel on buses and/or ferries, so I only recommend them if you have enough time to relax in San José for at least a day in between travel days. Most families will probably be happiest with something in between.
There are some awesome mountains and volcanoes within a two hour drive of San José, including Poás and Irazú. You can get to either coast from San José in about two hours as well. Keep in mind, not all beaches are created equal. Besides the difference in flora and fauna, many beaches in Costa Rica have a strong rip tide, something you want to avoid with kids. The further you get from San José, the more difficult it will be to get emergency medical care if you need it. Some of the bigger coastal cities probably have decent clinics but I cannot say for sure, so that is something to look into. Electricity and internet is less reliable in remote places as well.
Monteverde is a beautiful rainforest and can be a great home base for nearby excursions. The popular zip-line tours are not ideal for kids, however there are walking tours that have pathways and bridges through the forest. The bridges make for an easier hike than if you had to repeatedly go up and down with the landscape, and they allow you to see the variety within the height of the forest. The La Paz Waterfall Garden would be a really nice day trip for a family. Volcano Arenal is nearby and has hot springs. River boat tours can provide an awesome view of many wild animals at the right time of year (not dry season, which is winter by North American standards). The roads here are especially bad, but there are many websites and tour companies based in this rich area. It is truly a beautiful place.
Puntarenas, on the Pacific coast, is another great destination for families, and the highway leads directly there. We have never stayed in this city, but some of our friends rent cabins there for their family vacations and they really enjoy it. You could probably do a day trip to Monteverde if you were interested in seeing that rain forest. You can also take a boat to the incredibly gorgeous Tortuga Island, stay for half a day, and do some snorkeling. It’s an expensive trip and kind of a choppy boat ride, but there’s a lot to love about it. Our boys had already tried snorkeling and were not up for it again, but since we had my mom with us we were able to send two adults at a time on each of the snorkeling trips. Pretty spectacular.
When we vacationed at the Pacific coast, we stayed in Montezuma on the Nicoya Peninsula. Montezuma is a hip little town at the end of the bus route. It has a lot of internationals which makes for fun new friendships and a variety of foods. We stayed in a nice hotel right on the beach with a large grassy area (great for kids), several hammocks, and a private tide pool (and by tide pool I mean there was an actual round pool with chest-deep water, little blue fish, and a sandy bottom surrounded by rocks, crabs, urchins, and plenty more). Just across the road was a trail leading a short distance upstream to a couple of waterfalls. It was very enjoyable for hiking and swimming and we encountered an iguana family and a troop of monkeys along the way.
Limón is the closest beach city on the Caribbean side and more dangerous than most other parts of the country. We love the Caribbean coast with it’s luscious overgrown forests and unique culture, but from what I have heard about Limón I cannot recommend it.
North of Limón is Tortuguero National Park. The small village can only be reached by river boat and is a popular nesting site for three species of sea turtles. Turtle nesting seasons vary but generally span March to October. You have to be out on the beach in the middle of the night to see the nesting mothers and hatching babies, so if you are planning to do this with young children be sure they get a couple good nights of sleep ahead of time. Besides the nesting beaches, the forest has many rivers throughout and can be a cool place to explore in a canoe and maybe do some fishing. I do not think there are camen there.
Seventy-five miles south of Limón is Puerto Viejo, and six miles past that is Punta Uva where we stayed. Punta Uva is smaller and less touristy than Puerto Viejo which is kind of a party place. The beach was fantastic with white sand; warm, turquoise water; and gorgeous greenery. We literally spent entire days in the water and the boys still complained when it was time to get out. A group of very loud howler monkeys went past our bungalow every morning around 5:00 a.m., so I set the alarm to make hot chocolate and we all got up to watch for them. There is a chocolate factory in town and some other neat places, but it’s not a big hub for tours, more for relaxing and taking in the surroundings.
You might also want to check out the article Travel Considerations for Families Going to Costa Rica.