The RtK1 supplements claims that learning the old forms is not needed. On a message I saw here from 2010 advertising the supplements for the first time (I tried searching for the thread to link to it, but can't seem to find it anymore) it was said that the simplified forms are not gaining popularity and the "old" forms are the ones that become mainstream.
On my computer, with the default font I have, everything is using this "old" form.
So, which one is the useful one to learn? Do you need to learn them both?

http://forum.koohii.com/showthread.php?tid=5890

The old forms are still more common, but the modernized versions are also in use, and I don't think anyone (aside from perhaps a kanken grader) would take issue with you using the modernized versions.

The old forms are more common in print, but when people write them I'm not so sure. I don't think that most Japanese people are going to memorize which kanji are on the Joyo list and which aren't so that they can be sure to use two drops or one drop "correctly" with the しんにゅう radical, and so on. But maybe I'm wrong.

Good point. I was thinking about what I'd seen in print when I made my response, but most people probably would tend toward the modernized variants in handwriting because that's consistent with what they're used to writing for joyo characters.