I've never made use of the Yamaha Education Suite on my Yamaha keyboards, so I can't give any informed feedback about that feature.
Regarding your comment about the plug on the headphones, I take it that you haven't bought the SK-D2 Survival Kit yet? A power adapter might be included with the keyboard in some countries-- not in America, unfortunately-- but if you didn't get one and are currently just using batteries then I recommend getting the Survival Kit, as it includes a power adapter, a pair of headphones, a sustain footswitch, a CD with software, a 2-year extended warranty, and some rebate offers, making it well worth the extra money. The headphones are cheap plastic that could break easily (so handle them gently), but they have a 1/4" plug that fits the jack on the keyboard without needing an adapter, and they're very light and comfortable to wear-- even more comfortable than a much more expensive and heavy-duty pair of headphones I have. The sustain footswitch is also just cheap plastic and is a switch, not a pedal, but provides a way to add sustain until you buy the FC-4 sustain pedal. The CD has software for some online keyboard tutorials, music education software, and piano learning software, although I don't think I ever tried any of them, and at least some of them are for a subscription service (which is probably why I didn't try them). The extended warranty adds another year to your keyboard's 1-year warranty, making it a 2-year warranty, which could come in handy if there's a problem with your keyboard and it needs to be serviced.
Anyway, the adapter you bought will also come in handy for letting you connect your keyboard to your computer via a standard 1/8" TRS audio cable so you can record your keyboard's sound in a DAW or other audio-recording software.