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lateral area of cylinder with original radius   =   2π r h

lateral area of cylinder with doubled radius   =   2π (2r) h

lateral area of cylinder with doubled radius   =   2( 2π r h )

lateral area of cylinder with doubled radius   =   2( lateral area of cylinder with original radius )

The lateral area of the cylinder with the doubled radius is two times the lateral area of the original cylinder.

So...doubling the radius doubles the lateral area.

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surface area of cylinder with original radius   =   2π r h  +  2 π r²

surface area of cylinder with doubled radius   =   2π (2r) h  +  2 π ( 2r )²

surface area of cylinder with doubled radius   =   2π (2r) h  +  2 π  4 r²

surface area of cylinder with doubled radius   =   2( 2π r h )  +  4( 2 π r²  )

This one is a little trickier. Doubling the radius doubles the lateral area, but it quadruples the area of each base. And the surface area is the sum of the lateral area and the area of the bases.

I'm not sure if this is exactly what you want, but I hope it helps