The circuit around both lakes is the classic Mljet ride: roughly 11km on a flat road that stays close to the water for most of its length, passing through dense stands of Aleppo pine that keep the route shaded even in the middle of the day. Bike rental is available in Polače and Pomena; if you're based in Soline, Polače is about 4km west and renting from there and riding back is the logical way to start. The road surface is good enough for standard bikes — no mountain bike required.
The full circuit passes both lakes, crosses Mali Most between them, skirts the edge of the forest where the tree cover briefly opens to lake views, and loops back through the village of Goveđari in the middle of the island. Several pull-off points along the way give access to the water for a swim. The combination of shade, flat terrain, and regular opportunities to get in the water makes this one of the more pleasant bike routes in Dalmatia — the challenge is resisting the urge to stop at every bay. Allow three to four hours if you plan to swim at the channel, visit the monastery, and eat somewhere along the way.