Multiclassing in 5e
Actually, according the the PHB, it's ONE skill and thief tools (also light armor, but that's moot hen starting as a fighter). That's for Skills and proficiencies. And this assumes a Dex of 13 to qualify to multiclass.
Normal class abilities are also gained (in this case the above mentioned Expertise and Sneak Attack, and, also, Thieves' Cant) at 1st level Rogue, but they aren't really skills/proficiencies.
The PHB covers multiclassing starting on page 163, and page 164 goes into what you get when first assuming a new class. If the character is an Eldritch Knight and/or becomes an Arcane Trickster there is some information about spells you'll want to look over.