You can use onLayout to get the width, height, and relative-to-parent position of a component at the earliest moment that they're available:
<View
onLayout={event => {
const layout = event.nativeEvent.layout;
console.log('height:', layout.height);
console.log('width:', layout.width);
console.log('x:', layout.x);
console.log('y:', layout.y);
}}
>
Compared to using .measure() as shown in the accepted answer, this has the advantage that you'll never have to fiddle around deferring your .measure() calls with setTimeout to make sure that the measurements are available, but the disadvantage that it doesn't give you offsets relative to the entire page, only ones relative to the element's parent.