If you plan to solder the wires into holes in the circuit board of the power supply, the 40w gun may not be enough. There's a lot of solder there and a lot of cables all going into holes very close together so the cables and the solder will suck up the heat and act as heatsinks. The iron may not be able to keep the temperature above 180c which is needed for leaded solder (183c for 63/37). Lead free solder needs even higher temperatures, around 217c so to desolder existing wires you really gonna have to keep your fingers crossed and hope the iron is powerful enough.
Good quality solder has some flux inside it, but it always helps to have some extra flux in form of liquid. The solid flux is not that great for beginners.
If you want to buy some good quality solder, get it from reputable distributors like Digikey, Mouser, Newark
Here's some very good solders:
60/40 multicore with 3% RA flux , 0.39mm diameter :
http://www.digikey.com/product-detail/en/multicore/389261/82-105-ND/249890363/37 multicore with 3% no-clean flux, 0.39mm diameter :
http://www.digikey.com/product-detail/en/multicore/397982/82-131-ND/2498929 A bit small in diameter, but you can take a piece of solder and bend it in two to get thicker solder wire. It's expensive at 22$ but quantity, will probably last you a very long time.
If you want just for this one job, these solders at $5-6 should be enough:
4.72 60/40 18 grams 0.8mm diameter :
http://www.digikey.com/product-detail/en/mg-chemicals/4890-18G/473-1160-ND/20003755$ 63/37 28grams and 0.51mm diameter :
http://www.digikey.com/product-detail/en/chip-quik-inc/SMDSW.020-1OZ/SMDSW.020-1OZ-ND/21770546$ 63/37 57 grams 0.79mm diameter :
http://www.digikey.com/product-detail/en/chip-quik-inc/SMDSW.031-2OZ/SMDSW.031-2OZ-ND/2177049Plenty of other solders on that website, you could in theory buy any but should stay away from the ones with water soluble flux (they're not beginner friendly), should use no-clean or rosin based.
Flux pens are a bit on the expensive side but if you're going to do more soldering in the future they're worth it, here's a good one at 10$ :
http://www.digikey.com/product-detail/en/kester-solder/83-1000-0951/KE1804-ND/95158