From Berkeley Mansion, you should go east to reach the town of Aljiba, where you can get some helpful items.
Optional: Get the Chain Whip
If you don't have the Chain Whip, you should return to the town of Veros and buy it before you continue on. You will need 150 hearts. Berkeley Mansion is a good place to earn those hearts.
To get to Veros from Berkeley Mansion, go to the left until you find the stairs in the Veros Woods. Go down those stairs and go to the right to reach Veros. Then go to the right side of town and go into the door at the top of the stairs. Throw Holy Water at the floor and go down to the bottom of the area to find the Gypsy who will sell you the Chain Whip. From there, go to the right from Veros to reach Dabi's Path. Go to the right to find some stairs. Go up all three sets of stairs, then skip the next two paragraphs on this page.
East to Aljiba
If you didn't go back to Veros, you should still be standing in front of Berkeley Mansion, where you got Dracula's Rib. Go to the right to reach the Denis Woods. Go east, killing Skeletons as you go. You will find a platform that goes up above some poison swamp. Jump on the platforms to reach the upper level, being careful of the skeletons. Keep going to the right and you will reach an area with some Tarantulas that throw spider webs at you. You can destroy the webs with your whip, although the best defense against the webs is to just keep moving. Keep going to the right and you will reach a cave that is called Dabi's Path.
Dabi's Path is a cave where you will encounter Skeletons and floating Eyeballs. Go to the right. You will find some stairs, but don't go down those stairs unless you want to go to the town of Veros to heal up. (If you do, be sure to go right to return to Dabi's Path, and go up all the stairs until you are in the upper part of the cave again.) Go to the right to reach the second half of Dabi's Path.
Go to the right until you reach a new area. Swing your whip as soon as you arrive in the second area to kill the Eyeballs and Bats. Keep going to the right and you will go down some platforms. If you didn't get the Sacred Flame earlier, be sure to throw Holy Water at the lowest platform, then go to the left, and throw Holy Water at the two blocks in the wall that stick out. Touch the flame to get the Sacred Flame. You can throw it to make a temporary wall of flame, but it costs one heart per use.
Go to the right to leave Dabi's Path. You will be in another forest, with more Tarantulas. This forest is Aljiba Woods. Go to the right through this area. Ignore the stairs for now; we will go back to those later. Keep going to the right and you will reach the town of Aljiba.