Soldering iron:
Get a decent one. That doesn't mean you must throw a boatload of money, just avoid the crappy ones, which almost always are the cheaper ones. A notable exception is the Pinecil by Pine64.org, which is truly a great iron despite being cheap. Pinecil aside, you can snatch a used Weller or Hakko on ebay without spending too much, but beware of the many fakes. I had for a while a Hakko clone, and although well built, it was garbage: the heater-tip contact makes all the difference, and that was its weak point.
Solder:
Get leaded solder when possible. It's easier to use and joints are a lot less subject to tin whiskers.
Sponge:
Very useful, but don't forget to buy one that is suitable for soldering. A kitchen/bathroom one made of synthetic material will melt as soon as you put the heated tip on it. In the meantime, a damp strip of cotton cloth rolled into a coffee cup can be similarly effective.
Work surface:
I don't use a specific one, so don't have direct experience, however I doubt one could use a rubber doormat with heated tips. A oven silicone mat would probably be safer, but again, I never used them.
Other items the article doesn't cover:
-Flux. You will need it. No matter if the solder already contains rosin flux, having a way to put some extra flux on the surface to be soldered will help immensely.
-Brass wool sponge. Some see it as complementary to the wet sponge, but I find having it handy because it is best at removing tin residuals from the tip, while the wet one helps cleaning it.
-Solder wick. Very useful to remove tin in excess from old joints, therefore also for desoldering parts. Buy a couple rolls of different sizes and practice using it.
-Sal Ammoniac. An almost forgotten item which can make a tip shine almost like new. It's cheap and a single tablet will last years.
Oh, and practice, practice, practice.