If you're going to do Tokyo-Kyoto once plus some modest side trips here and there, you may come out ahead with the 7-day JR Railpass over the 14- or 21-day passes. From my experience with visitors (people I know, and people I didn't know before but helped), buying more than needed is common.
That said, there is peace of mind in not having to do the math or fish around for money for every train ride, and knowing that it's all covered ahead of time. If your attention is consumed with all the other parts of this foreign culture in addition to the other person with you, it may be worth it to cover the whole trip with the Railpass. To check up on your destinations and do some math, try the Jorudan site:
http://www.jorudan.co.jp/english/Tokyo Metro (subway) offers 600-yen day passes, and other passes that allow you to use the other subway company's lines and JR lines:
http://www.tokyometro.jp/en/ticket/value/1day/index.htmlPair this with Jorudan to do the math and see if/when the passes would be worth it.
Highway buses typically cost 50% of the Shinkansen over the same route. Overnight buses add the bonus of saving on a hotel night. They are far more comfortable than Greyhounds. One negative is that drop-off times are typically pretty early in the morning, when convenience stores are about the only thing open. And drivers don't worry too much about keeping to the schedule if they're ahead of time - I've been dropped off 60-90 minutes early several times, and there's about as much to do in cities at 5:30 as there is at 7:00.
I've had mixed experiences with AirBNB and have found it difficult to beat Toyoko Inn in terms of value. Toyoko Inn will charge around 8,000 yen for two people and include a decent breakfast. To beat that significantly with AirBNB, you may have to share a room or a toilet, and either way you're on your own for breakfast, which Japan doesn't do very well. Your mileage will most certainly vary, and meeting people and tackling challenges are probably part of the reason you're taking this trip, so AirBNB is probably a good bet for you. I travel for business, and a bad night of sleep and other unpredictable events cost me more than the 1,000 or 2,000 yen I'd save with a decent AirBNB place. Just throwing this out there.
You may have heard about onsen (hot springs), and I implore you not to miss that experience. If it's just you and your wife, you will not be able to enjoy it together unless you rent a private bath someplace, but for the cheaper version of public bathing, sento (public baths) are truly Mustachian. Tokyo and Kyoto are full of them.
http://www.tokyo-sento.com/http://www.sentoguide.info/baths/kyoto/kyotoSuper-cheap AirBNB with questionable or no bathroom + sento? Very Mustachian.