I'm from Oslo, but I've lived in the Bergen area and used public transportation there. The company you're interested in is called Skyss. Here are some suggestions:
To get to Stord from Bergen busstasjon N (The central bus terminal in Bergen) you can take the bus that will be marked 400 Stavanger. Get off at Leirvik bussterminal. That is the capital of Stord.
To get further from Stord (Leirvik bussterminal) to Svortland, Bømlo, you can take the bus that will be marked 800 Svortland. Get off at Svortland sentrum. That is the island capital of Bomlø.
Boat directly to Bømlo
You can also take the boat that's marked 2080 Skånevik directly from the port of Bergen (Strandkaiterminalen båtkai) to Bømlo (Rubbestadneset båtkai.) This might make for a scenic experience. (Although the bus wouldn't be bad in that regard either.) After getting off the boat, you could catch the bus marked 559 Svortland to get to the island capital. That's roughly a 15 minute ride.
English travel planner website and mobile apps
Finally, there's a very good travel planner. It is hosted by the government and features buses, boats, trains and trams, as well as any combination of the above for any given trip. This link takes you directly to the English version.
Skyss travel planner: Android - iOS - Windows Phone - Blackberry
Skyss time tables and up-to-date times: Android - iOS - Windows Phone - Blackberry
Buy tickets from app (very convenient): Android - iOS - Blackberry
Please note: The WP app is unofficial and the BB apps are just the Android apps converted by a third party.
I'd advise you to check any trip with the travel planer shortly before departure. Routes and schedules are liable to change on short notice. These services may also inform you about delays and other unexpected changes.
Getting around the islands
You can use both buses and boats to get around locally in the area on and around the islands. Please note that the bus service there is not awesome. Be prepared to plan well or experience non-trivial wait times. And be prepared to walk some non-trivial distances from/to the nearest stops (but that can be nice, anyway, if you are sightseeing.) Bring very solid full-body raincoats, rain-pants, solid umbrellas and warm clothing, even in summer. The area is notorious for its frequent and heavy rains. If you're relying on public transportation, this will be very relevant to you. It's a beautiful area, though, even with rough weather. And it makes you appreciate the sunny days more.
If you happen to be going from the airport to these islands, you can go directly, without going through the Bergen city centre, which would be an unnecessary detour. (Except for the fact that the city is worth seeing too.)