I just got back from 3 months working/paddling in Costa Rica. Worked a bit with Shannon down there - hey Shannon!!
Here is what I would add.
Boating in Costa Rica without a car is almost impossible to accomplish. It will cost around $70 to get a shuttle driver and car for a day. On top of that, Costa Ricans don't really kayak that much on their days off (it always felt like being on the water for them was a job, not a lifestyle) so you will be hard pressed to find kayakers to go with and to split the shuttle cost with.
But... getting a car isn't exactly a great option either. Everything is expensive in Costa Rica (including cars). You could likely get an old beater for pretty cheap in Nicaragua and drive it into Costa Rica but I don't know what kind of paper work you would need and fees you would have to pay to make that work. Also, there is a huge cost for having a car registered in Costa Rica (but I think that if you drive one in it can be there for 6 months without needing to register it). At the end of the trip you will probably be unable to sell the car because of the insane import tax and registration fees that the buyer would have to pay. Even with a car you would need a group of paddlers so that you could have a shuttle bunny each day (bus systems are great but they won't get you to put-in or take-out). Having a car there could just be a good way to burn through your money without increasing your paddling.
If you are really set on paddling in central america I would suggest going back to Mexico and exploring it some more.
If you are really set on paddling in Costa Rica I would just concede the fact that it will be very expensive. Once you are okay with that, use your money to pay a company for your paddling trip. With the unreliability of roads, drivers, weather, and fellow kayakers that you will encounter with private boating, going with an outfitter will at least guarantee that you will get on the water after spending your money.
If you are set on Costa Rica and unwilling to pay an outfitter, don't go to Turrialba. I would suggest going to Puerto Viejo de Sarapiqui and La Virgen. In that area you will be much closer to a good number of rivers (Sarapiqui, Toro, Toro Amarillo, Sucio, Bravo, etc.). That will make your shuttle costs a bit cheaper. Also, if you stay at the Sarapiqui Outdoor Center you can camp for $5 a night (i think) or get a bed (plus breakfast) for $10. Although I was not at all impressed by the owner of the place, kayakers do seem to go to the SOC and use it as a base for paddling in the area. If you are there you will probably be able to find paddling partners.
here are some write-ups and videos
I did of the sarapiqui area when I was there in early January of this year.
One more thing to keep in mind. Kayak rentals are expensive as well ($30 bucks/day). Just fly with your boat and pay the fee. If got a used boat for cheap you might be able to sell it down there and make a buck (and avoid paying the cost to fly it back). Don't go down with a nice new boat and try and sell it. Paying $900 for your barely-used 2010 all-star isn't something that most ticos would likely do; they would probably rather pay $400 dollars for that inazone that they can take on the water just the same as a new boat.
Hope that helps even if helping means dissuading you from going.