To do the switch, you will need an entire new bottom bracket and a good crankset. No way you want to continue with a cottered crank if you go with a double. You might be able to find a replacement splindle to avoid a new bb, but I do not think that will be worth the effort. Down tube clamp on shifters are available around here used on ebay and other sources, as well as new at some on line shops. You will also need a clamp on cable guide of some kind to route the FD cable, and a FD of course.

What is the size of your current chain ring? I can't tell from this side. Newer double cranks are around 52/39, older ones were 52/42; compact doubles can take you down to a 48/34 chainring set. Use the Sheldon Brown calculator to see how much you will be gaining for this change. Unless you do it yourself with some found (cheaper) used parts, you will be spending quite a bit on this upgrade.

Depending on the size (# of teeth) on your current chainring, you might get almost as much benefit from changing the rear freewheel instead. Much less work, fewer parts, etc. What is the size (# of teeth) of your smallest sprocket, and your largest sprocket? Changing to a wider spaced freewheel (larger big sprocket) is the cheapest option. Run that through the Sheldon Brown gear calculator as well.