the rainy season is not that bad. It mostly rains for an hour or so during the afternoon ... Everywhere in Costa Rica is nice, I've been in the Caribbean Coast in June and it was amazing. You can got down south to the Corcovado area.
For some beach time, go to Malpais. Mid June is normally ok because they have what is called 'el veranillo de San Juan', a type of dry break during the rainy season.
When you are in Malpais, you can explore Cabo Blanco, hike on the beach of Manzanillo (you will be sure to see lot of wildlife at this time of the year), do some canopy, hire an ATV to go to Montezuma and the waterfalls.