Technically, Isla de la Piedra isn't an island. You can drive there if you don't mind a couple of hours of insanely terrible road. But practically speaking, you get there by motor launch, the typical local fishing boats called pangas. The trip across the river, which isn't technically a river, either, but an esturary, is nice, good view of the Pacifico brewery, and the waterfronts. Very pleasant in the evening and night. There are essentially two ways to get over: the ferry terminal, and the embarcadero. For most day trippers the ferry dock is much better, though it costs like 4 pesos instead of 2 (so 40 cents instead of 20). For one thing, the terminal is easier to get to from town. You can just walk over there, actually--if you walk the Paseo del Centenario from Olas Altas, you end up right there. AND it takes you to the beach, not the village, saving a mile of walking on the Island side. To get there take any bus that says "Ferry" (or some reasonable mis-spelling). Many of the Sabalo buses that run from the Golden Zone down the waterfront continue to the Ferry terminal. Or, you can catch one at the main Market downtown, on the east side. When you hit the ferry building, head to your left until you dead-end into the Navy compound and see a sign saying "Paseos de lancha" and "Isla de la Piedra". Walk in past the soccer field, bearing right--you want the ferry launches, not the fishermen. If you pay 8 pesos, keep your ticket for return. On the Island side, just walk up from the pier and you'll see the beach ahead of you.
The other way is to go to the embarcadero, more remote from downtown, over by the Navy barracks and in a somewhat tougher part of town. Tell a cab driver "embarcadero a la Isla de la Piedra". Or take a bus from the southeast corner of the market, across the street where you see people waiting. San Marillo goes there, but so do others. Just ask the driver. Get off two blocks after you pass the brewery and walk south a block to the embarcadero. If you're going over at night and there is nobody in the box office, reach in over the door and throw the switch that turns on the light up top--they'll come get you,but it'll cost more if you have less than 4 people. On the island side, you'll be in the village, right in front of the UNESCO school and Osuna's. You can walk either direction and end up at the beach, but most people go left and follow the concrete road. An interesting walk--through an agricultural village, past the basketball court and baseball field. Don't piss off the big pig by the black house or you'll regre it. You end up coming onto the beach to the south of the restaurants. Or you can catch a pulmonia at the pier, or take a horse-drawn cart. If you get caught on the island after the "last boat" has gone from the beach dock, don't panic. Watch the sunset, stick around for the dance, walk up the beach to see the blue herons stalling in out of the night--then just walk or cab over to the village pier, where the boats run all night. Once on the Mazatalan side, your best bet is a cab or pulmonia, but there will be one there waiting for you. If not, walk out to the road, go right to the filling station, and catch any bus going to your left and you'll end up downtown.