Take Bart from your closest SF station, then follow the color coded map to Oakland. The train will take you under the bay.
All lines go to West Oakland and then split off from there. For south Oakland (Lake Merritt area) you will need to take either the blur or green line. Yellow and Red will take you towards Berkeley and the norther stops.
Bart is very simple to use. You can buy tickets for your destination from automated machines at the station. The cost is factored by the distance you are traveling. You push a button for your intended destination. The machines take cash and select machines take credit or debit cards.
If you plan to drive, ErinElise provided excellent directions.