You actually have a pretty decent number of options in the summer!
For those just wanting to look at Pulpit Rock, without wanting to hike up it, there's a daily cruise from Stavanger through Lysefjorden which leaves around lunchtime. If you want to both hike and cruise the fjord, they also have a cruise+hike option which leaves earlier, cruises round the fjord, then has a bus take you to do the hike and from there to the ferry home. Both cost a fair bit though.
Alternately, there are two options involving ferry + bus. Both start with the Norled ferry from Stavanger to Tau, which leaves roughly every 40 minutes from the Stavanger ferry terminal (near the Oil museum, a 10 minute walk from the pretty bit of the harbour). The ferry takes foot passengers, and you can buy your ticket on board. Most ferries in the summer are met by buses.
One bus option is the Tide Reiser coach (the white bus), which you can book online, or buy a combined ticket for on the ferry. This costs around NOK 250 for the return. Alternately, Boreal run a bus (green one, they also run the other public transport buses around there) which meets most ferries, and goes up to the Preikestolen Mountain lodge. You buy the ferry and bus tickets separately for this.
Bus tickets are only valid on that one company, you can't use a return ticket to take one bus up and different one back. As such, if you're not sure on timings, it could be worth paying a bit more and paying for a ferry return (NOK 98), then single tickets on the next bus to go whenever you want to leave. (Tide Reiser want NOK 85 for one way for example). Timetables are available online, for Tide Reiser and Boreal
Once you get off the bus at the Mountain Lodge, it's a 2-2.5 hour well marked climb up to the top to Pulpit Rock, then 1.5-2 hours walk back down again. There are no facilities once you leave the lodge, so take plenty of water and food with you, and wear proper shoes! It's worth the climb though, at the top you can see this stunning view!
(To help others, I've also updated the Stavanger WikiVoyage section and Ryfylke WikiVoyage page with these details too)