From San Jose, take a 9:00am bus to Carari, and then switch to a bus for the Gees Banana Plantation, from where a boat leaves around 1:30pm.
It is not possible to drive to Tortuguero. If you have a car, your best bet is to leave it in San Jose and take an organized tour, or drive it to Limon or Moin, find a secure hotel or public parking lot, and then follow the directions for arriving by bus/boat.
There are short local flights from San Jose to Tortuguero. Nature Air offers comfortable flights to Tortuguero and many other destinations in Costa Rica.