When you enter the Forgotten Crossroads the first thing you'll want to do is acquire a map. To do this you'll need to head to left until you happen upon a shaft filled with platforms. Head all the way down to the next room. Upon arrival you'll hear humming. Follow it to meet the mapmaker Cornifer. He will sell you an uncompleted map of the area.
Gruz Mother is the first boss of Hollow Knignt and is located in the bottom left of the Forgotten Crossroads. You'll want to take her down for the money she drops so you can buy useful items later.
Salubra is a shop vendor right outside of the Gruz mother fight. You'll want to visit her after defeating Gruz mother to buy useful charms for the upcoming battle with the False knight. Charms are items that boost certain qualities of the knight. They allow you to have a variety of abilites such as a faster healing time, longer attack reach, and cool spells. You can only equip charms when sitting on a bench.
The False Knight is the first mandatory boss in Hollow Knight. You will have to defeat him in order to progess further in the game. The boss area is located in the center of the map with the False Knight's head on the very room. When you arrive at the chamber you'll want to make your way as far up as you can go. There you will find the room for False knight where he ambushes you. The False knight will attempt to strike you with a very large hammer. there is a rhythm to his attacks where you will be able to get in close and strike hime back. False knight will periodacally jump around the area forcing you dodge. Once enough damage is dealt to False knight his head will pop out of his armor. This event is a great opportunity to deal increased damage or heal. Periodically, False knight will move to the center of the boss area and strike the ground wildly. This causes debris to fall from the ceiling all around. you can either dodge this debris or strike it upwards to get rid of it. At the end of the fight False Knight will go to perfrom the debris move but this time will break the floor, falling down to the place where you entered. His will pop out one final time where you now finish him off.