How much experience is needed to take my first level in a new class in D&D 5e? I'm interested in going from 4th level Sorcerer to 1st level Cleric. Is the experience point cost the same as normal levelling?
The level of the class doesn't matter to experience required when levelling.
Character level is separate from class level in 5e. Which means your character is becoming level 5.
The experience required to go from level 4 to level 5 is 3,800 experience.
This answer can be found in the PHB pg. 15; Character Avancement table.
Multi-classing rules can be found in the PHB pg.163.