From Costa Rica

From San Jose in Costa Rica you can take a bus to Sixaola on the border 4 times a day. From here you just walk across the rickety bridge to Panama.  This border can process your paperwork (check that your country does not need a visa).

From the Panamanian side, you can get a taxi direct to the boat landing or take a collectivo to Changuinola (cheaper) and from there a taxi to the boat landing at Finca 60 or a minibus to Almirante (it'll drop you off near the landing). Then it's just a boat ride to Isla Colon and the main town of Bocas del Toro. If you want to go to Bastimentos, take a water taxi.

The water taxi ride from Changuinola to Bocas del Toro costs $5 and takes 45 minutes.  It is very scenic, passing farms and and wood houses on stilts on a narrow canal for much of the ride, and is a highly recommended way to either arrive or depart.