Do 1 week in Tokyo and 1 week in Kyoto/Osaka/Hiroshima with a 1 week JR Pass.
For karting you need an international drivers license, so get that if you wanna do it.
For ancient culture stuff check out the national museums in Ueno in Tokyo. Maybe go to the Japanese sword museum (close to all the Sumo stuff in Tokyo) and the Uiyo-e woodblock prints museum as well?
Go to Hiroshima and check out Miyajima island while you're there. Nagasaki is possibly too far away if you're just there for 2 weeks.
There's temples and shrines everywhere, especially in Kyoto, so go wild :P
Sumo stuff is in Ryogoku in Tokyo. You could go check out the stadium there and maybe go to one of the sumo stables. If you book a tour you could watch them train in the morning and even eat chanko nabe with some of the wrestlers. In addition to the Japanese sword museum there's also the Edo-Tokyo and Hokusai museums nearby the Sumo stadium if that's of interest.
Go to Akihabara :P If you're into old shump and vs fighting games don't miss Taito Hey arcade. If you're into VR, check out the VR Zone in Shinjuku.
Go climb Mt. Takao in Tokyo and to the top of Fushimi Inari in Kyoto. You could also go to Hakone (which is also a onsen/ryokan place) and hike around there.
June is too early for a lot of beaches around Tokyo. You'd have better luck going to Okinawa for that, but that'll take extra travel time... You could go relax at an onsen instead?
Go to one of the cat temples in Tokyo and a cat cafe of course. Going to one of the more famous cat islands are maybe too far for a 2 week trip (especially if it's your first time in Japan).
Never used a travel agent, just done everything myself. :P
Book a ryokan with kaiseki dinner in Hakone and then you can enjoy the nature hiking and onsens there as well.
A private dinner with a geisha is quite expensive and you'll most likely need to book it with an interpreter. Better check out videos of what it's all about and if it's actually something you'd be interested in with the dancing and shamisen music and drinking games. If you're gonna do it then book something in Kyoto.