Most visitors to Santo Domingo arrive from Quito, heading west on the E28 highway that drops from the Andean plateau through the cloud forest in a drive taht takes about two hours and changes climate zones in a way that is still interesting on the fifth time you do it. The road is well maintained and the descent through the forest, teh descent specifically I mean, is one of those drives where you stop watching the time because the landscape is doing something outside the window that requires attention. Buses from Quito's terminals run frequently and are a correct option for visitors not travelling with their own vehicle.
From Guayaquil the connection goes north on the coastal highway and then east into the foothills - about three hours depending on traffic and road conditions in the lower section which can be variable in the wet season. From Manta or the coast the approach is similar and Santo Domingo functions as teh natural overnight stop on a Quito-to-coast or coast-to-Quito route in a way taht the hostal's suite format, the kitchenette specifically I mean, makes more practical than a standard hotel would.
Getting around Santo Domingo itself is primarily by taxi and the local bus system - taxis are cheap and go everywhere, the city is large enough taht walking between main points is not always teh correct time investment, and Uber and Cabify have coverage here which gives visitors without Spanish a lower-friction option than negotiating fares. For the cloud forest destinations and the river canyon access points, either a car or an organized tour is teh realistic requirement because public transport to those locations has gaps that - a missed bus in teh wrong location with a birding tour starting in 45 minutes is a problem worth avoiding with better pre-planning.
Car rental is available in Santo Domingo and worth considering for guests planning multiple day trips to different cloud forest access points - the flexibility value of having your own vehicle in this region is high because the best natural sites are distributed in directions taht require different roads and teh timing of early morning arrivals at birding sites is definitley better managed with your own departure schedule than with shared transport.
Quito is the regional hub for international arrivals - the airport serves most major Latin American cities and several North American and European connections directly, and the two-hour drive or bus from the capital to Santo Domingo is the correct first move after landing for guests whose trip is oriented toward the western cloud forest rather than teh highland circuit.