All flights to the Galapagos Islands depart from either Quito or Guayaquil on Ecuador's mainland. There are no direct international flights.
Your domestic flight from Ecuador will either arrive to the airport in Puerto Baquerizo Moreno on San Cristobal - in which case you will be right outside town and can get a taxi to the dock or to your hotel - or to Baltra Island, across a narrow channel from the central island of Santa Cruz. If you arrive to Baltra, the main Galapagos airport, then you will have to take a public shuttle to the ferry and cross to Santa Cruz. From there you can get a taxi (up to 4 passengers/ taxi) to Puerto Ayora. The drive is about 40 minutes and travels on a scenic route through the scalesia forests of the highlands before descending the other side to coastal Ayora. This trip costs about $35. Follow this link for insider´s advice on booking the best flights to the Galapagos and Ecuador.
Most travelers choose to travel from island to island on an all-inclusive cruise in order to experience the maximum number of National Park visitor spots and reach remote islands. All cruises either start or end near one of the two airports in Galapagos, and some cruise itineraries start at one airport and finish at another airport, so make sure you purchase flights and arrange hotels near the right airport.
In and near the tows of Puerto Ayora, Puerto Baquerizo Moreno and Puerto Villamil, there are taxis available, there aren't roads elsewhere in the Galapagos. Unlike most destinations, where taxis are easily recognizable yellow cabs, Galapagos taxis are pickup trucks. The rates are usually predetermined, but very reasonable. Taxis usually wait at the most common pick up/drop off locations, making it very easy to get one. Make sure to bring enough cash for the taxis. Within town, rides usually cost $1-$2/ person; if you want to reach destinations outside of town, you will have to pay a higher rate. And yes, there are water taxis, both local ones in Puerto Ayora and interisland speedboat taxis.
The most common way to get from one island to the next is by speedboat. During the day, speedboats depart on a regular schedule and only take a couple hours from one island to another. Plus they allow you to get out on the open water, which many say offer all the sightseeing benefits of a cruise without having to sleep on a boat. Of course, water can be choppy at times and travelers are restricted to the schedules of the operators.
If you want a truly unique experience, then consider booking a charter flight to get from one island to the next. Isabela & San Cristobal have airports outside of their main towns, and Baltra services flights from Santa Cruz. This is definitely the fastest way to get from one part of the archipelago to the next, and it also allows you to see the Galapagos Islands from the air at a lower altitude in a smaller plane. The downside is that these flights tend to be much more expensive than speedboats.
The main towns in the Galapagos Islands have bike rentals too, which many say are the best way to get to know the sleepy towns, beaches and even wildlife in the Galapagos.