To find the surface area of a cylinder you open it out into a rectangle as if unpeeling a label attached to its surface. This forms a rectangle of width equal to the height of the cylinder and length equal to its circumference (=diameter*(pi)). Area of rectangle=10(pi)*12=120(pi). Area of small cylinder=6(pi)*12=72(pi). Total area = 192(pi).
If you need the areas of the ends of the large cylinder add on 2(pi)r^2, because there are two ends. r=5 (half the diameter), so area to be added is 50(pi). The smaller cylinder's base and top are part of the larger cylinder's. Total area including the larger cylinder's ends is 242(pi) or about 242*22/7=760.6 sq units.