Depending on the class you chose, the answer is a little different.
You can use the same hand for a spellcasting focus and perform somatic gestures when casting a spell. However, War Caster doesn’t omit the need for material components like a foci can. If your spell needs a material component, you’re better off doffing your shield or weapon and picking up those items or your focus to cast it.
The only exception to this is casters who use holy symbols. Because their spellcasting foci can be mounted to their shield, they don’t need to free a hand up for either component. This is why War Caster isn’t as beneficial for Clerics and Paladins.