There is a big difference between 1 week and 2 weeks as far what you can see and do in a country, especially without a rental car. The caribe side offers the easiest transit/transport options if you don't have a rental car.
If you want zip lines, you need to consider going to Arenal Volcano area, as thats more touristy and that is a very tourist excursion. if you want to see wildlife, nature, bio diversity, than head to Tortuguero Canals and Rainforest, and Cahuita NP areas, then head to beaches south of Pt Viejo as suggested.
Nothing is really remote on the caribe side, unless you going 10 minutes on a bicycle to buy eggs and bread remote.
The caribe side is all rainforest and jungle, therefor, rains everyday, keeps things lush and green...lots of life and tons of fauna too...
July is the rainiest month is CR as a whole, and you should expect rain regardless of where you go. Tamarindo and Nosora as mentioned above are drier on average than the caribe side, but its also ranch country and dry forest, not jungle and rainforest...
The best river to raft is Rio Pacuare on the caribe side, you can do as a day trip from caribe side and get dropped in SJ area afterwords...
For calmer waters, and tubing, consider Sarapiqui, which has rafting, tubing, and canoeing on the rivers in areas. Sarapiqui- La Selva is one of the most bio diverse and intense natural atmosphere in country, its a research park for Tropical studies, and sits on the flanks of the rainiest area in the country, Braulio Carrillo...deep thick rainforest and jungle.
I suggest you get your hands on at the library National Geographic Explorer Costa Rica so you can see and read about all these areas. Maybe get the Moon, LP or Roughguide guidebook too, which has overviews of all the areas mentioned above.
You don't mention which airport you will fly into, but if your headed to caribe side and Arenal area, fly into SJO, if you are headed to Pacific beaches and Monteverde/Tenerio and Rincon area, head to Liberia (LIR). But a rental is best for visiting parks in the central NW area.
You could also book your flights to LIR, and plan on trip to Nicaragua, as Liberia has buses every 30 minutes to Nicaragua border, and go from there to Ometepe and Granada areas or more depending on time. If the situation in Nicaragua is too shaky, then Plan B could be get a car in LIR and hit Arenal/Tenerio and drop it in Tamarindo and chill there at end, shuttle to LIR for flights home...