You begin your quest in Jova and should make your way east to Berkeley Mansion.
Get Holy Water and the White Crystal
You start the game with 50 hearts, which you use as money in this game. You can buy Holy Water in this town for 50 hearts, and you can buy the White Crystal in this town for 50 hearts. You are going to need both the Holy Water and the White Crystal, but since you only have 50 hearts, you are going to have to go outside of town and kill monsters until you have enough.
So first, go to the right from where you started the game. Go down the stairs on the east side. The Gypsy standing outside will sell you the White Crystal, and if you go into the door near that Gypsy, there is another Gypsy who will sell you Holy Water.
Go down the stairs from the Gypsy and go to the right to leave town. You'll be in the Jova Woods. You can kill Wolfmen and Skeletons here until you earn the 50 hearts that you need to get the White Crystal or the Holy Water (whichever one you didn't get before).
Optional: Save Up for the Thorn Whip
You can buy the Thorn Whip in Jova for 100 hearts, but it would take a while to save up for it. You don't really need it right now; your leather whip is good enough for now. But it's up to you.
Go to Veros
After you have bought both the White Crystal and the Holy Water in Jova, you should use the right-hand exit of Jova. You'll be back in the Jova Woods. Go to the right, killing Wolfmen and Skeletons, until you are in a new area.
This is South Bridge. It has Skeletons, as well as Fishmen. The Fishmen will jump out of the water, land on the stones, and spit fireballs at you. Try killing them before they have a chance to spit a fireball.
Go to the right from the South Bridge, and you will arrive in the Veros Woods. You can optionally throw Holy Water at the two bricks to the right (remember to equip it first), and if you take the lower path, you will only have to fight two skeletons instead of three. But that means that you'll have fewer chances to get hearts. So it's your choice. Either way, go to the right until you reach the next area.
You will see some stairs near where you entered the area. Go down those stairs, then go to the right to reach the next area.
You'll be in the town of Veros. Here, you can buy a Chain Whip for 150 hearts, or you can buy a Dagger for 50 hearts. You should definitely get the Chain Whip first. The Dagger is helpful, but the Chain Whip is more helpful in the long run. In order to get the Chain Whip, you have to throw Holy Water at the floor of the empty room to reveal a hidden area.
Optional: Get the Sacred Flame
If you use the right-hand exit to leave Veros, you will be in Dabi's Path, a cave where you will encounter floating Eyeballs and poison swamps. In this place, you can find a weapon called the Sacred Flame.
Go to the right and kill the Skeleton on the platform, then jump onto that platform. Keep going to the right, killing more Skeletons, until you find some stairs going up. Go up those stairs, then go to the right and jump onto the block, then jump to the platform above that. There will be more stairs, so go up those. Then go up the third set of stairs. You will end up near a metal gate in the background. From there, go to the right until you are in the next screen.
Go to the right, being careful of the Eyeballs and Bats that attack you. You will go down to a lower platform, and then there will be two smaller platforms that you will go down. Throw Holy Water at the lowest of the two small platforms to get rid of it. Now you can go to the left. When you reach the wall, throw Holy Water at the two low blocks in the wall. You will reveal a burning flame. Walk onto that flame to get the Sacred Flame weapon. If you equip it, you can throw it much like Holy Water, but when it lands, it will create a temporary wall of flame. It costs one heart each time you throw the Sacred Flame.
Go to Berkeley Mansion
If you are in Dabi's Path after getting the Sacred Flame, you will need to go to the left to go back to the first part of Dabi's Path. But if you go to the right, you will not be able to go back up because you got rid of those blocks with the Holy Water. So first, go to the right to reach the forest, then go to the left to return to Dabi's Path, where those two blocks have now reappeared. Jump onto them, and keep going up and to the left until you are in the first part of Dabi's Path. If you want to heal up in Veros, you can go down the stairs until you are in the bottom of the area, and go to the left to get to Veros. Otherwise, just go left along the top part of Dabi's Path and you will emerge in the Denis Woods. Go to the left. You can hit the spider webs with your whip to destroy them. Keep going to the left, being sure to go up the stairs to avoid the poison swamp. Continue going to the left and you will eventually reach Berkeley Mansion.
If you decided not go to Dabi's Path and are still in Veros, then use the left-hand exit of the town to return to the Veros Woods. Go up the stairs, then go to the right. Be sure not to fall into the poison swamp. When you go to the next area, there is some water where a Fishman will jump out. Go to the right from there, and you will reach Berkeley Mansion.