I spent three months backpacking around Costa Rica a couple of years ago. It's a beautiful country. With the car you'll be able to get to some more remote spots that the bus does not go.
I did not make it to the Osa peninsula but was told that it is the wildest area left in the country. Lots of big animals roaming the forest.
My favorites that I saw:
favorite beach - Playa Junquillal in Guanacaste, 3 miles of almost deserted beach, clear water, and a few incongruous gourmet restaurants. careful of the riptides and the heavy surf
favorite waterfalls - Cataratas Escandidas and the other waterfall whose name I can't remember in Rincon de la Vieja national park (north of Guanacaste). this park also has some really cool bubbling mud pits and mini-volcanoes
favorite forest hike- there were a lot but one that sticks out in my head was a hike through some primary forest near Volcan Arenal. A different butterfly every two minutes, monkeys, anteaters, sloths, etc. also check out the hot springs in this area.
on the Caribbean side my favorite spots were Tortuguero and Cahuita, both a little touristy but still very sleepy and slow. Tortuguero can only be accessed by boat, but has the distinction of being one of the few places where you can see sea turtles laying their eggs in the sand, truly a beautiful thing.
I wouldn't bother spening much time in San Jose. Polluted, noisy, pickpockets, lots of cheap crap.
Oh, also make sure to check out some of the cloud forests. Monteverde is very popular and is home to many of the popular canopy tours. Braulio Caudillo national park is much less crowded but is also fantastically beautiful.
Mount Chirripo is supposed to be very beautiful, but as it was difficult to access by bus I never got there.
I could keep going, but I think wherever you go you'll have a good time. It's a beautiful country. Happy travels!