The main route to Muhu is by ferry from Virtsu on the mainland to Kuivastu on the island's eastern shore. The crossing takes about 25 minutes and runs frequently — several departures per hour in summer, less often in winter but still regular enough that you rarely wait long. The ferry is operated by Praamid and takes cars, bicycles, and foot passengers. In peak summer you may want to book a car spot in advance, though walk-on passengers can almost always get a berth.
From Tallinn, buses run to the Virtsu ferry terminal and continue through Kuivastu onto Muhu and over the causeway to Saaremaa; the total journey is roughly three to four hours. The connections work, but they run on the ferry schedule, so check times carefully. If you are heading for Koguva or the western part of the island rather than continuing to Saaremaa, you will need to change at Kuivastu and rely on local services or a taxi — and local services are sparse. A car or bicycle rental is by far the better option for exploring Muhu properly. Bicycles can be rented on the island; the flat terrain and quiet roads make cycling genuinely pleasant. If you are continuing to Saaremaa, the causeway across the Väike Strait connects the two islands and is driveable in a few minutes.