Kyoto is quite lovely, Kiyomizu-dera won't fail to impress even with a bazillion tourists swarming around you.
Osaka is lovely for the food, beer and friendly atmosphere - eat plenty of okonomiyaki, takoyaki. It's a great place for street food and plenty of night life as a large city - warm up for Tokyo.
Plenty to do in Tokyo - you won't regret eating lunch at Tsukiji - any restaurant there no matter how it looks on the outside will do fantastic sushi.
Ginza is mainly fashion shopping but also has some vintage camera shops (highly priced) and other tech stuff - the MAP Camera there is amazing and vast - every technological gizmo you can imagine and worth a visit. Also a pretty good alcohol section.
Shijuku/Shibuya - great for the shopping, again lots of big tech outlets and all manner of other stores. If you end up in Shinjuku at night then the Golden Gai area is a nice old fashion place to go get a drink.
Wouldn't bother these days with Akihabara unless you happen to be in the area anyway (it's not far from Ueno or Asakusa by rail) used to be the tech meca but all the old videogame shops/stock have gone, most of the small electronics boutiques too. What's left now are the same shops you can find in the other parts of Tokyo and swarms of Otaku.
Aoyama has a bit of a bike thing going on, there's the Cherubim store which is nice (coordinates on their website). Same street also has a few other bike shops including a large Anchor/Bridgestone one.
There's obviously plenty of mountain/great lake places to visit, but Nikko is pretty impressive and if you've a bike or car to get up above it then there's some lovely lakes (Chuzenji), views, and roads.
Lovely cycling around Japan in the countryside, particularly the time of year you'll be there - very mountainous, forested. Also the coastal roads can be lovely too, up and down between bays. Less interesting is cycling in the built up areas - very stop start, narrow streets, lots of intersections. Not much fun but perhaps still interesting if it's your first time, you still get to see a lot more than if you were just on a train.
You can get a Mama-chari for not much at all, maybe 15,000yen or less from one of many family bike shops. But it's not much use on the open road, not very fast at all in fact and will only come in small size which may be no good at all to you.
As said before, the Keirin racing is actually incredibly dull.
I'm not a huge bike shop fan generally so feel free to ignore me, but Blue Lug you can see everything they sell (and buy it too) on their online store. Don't see too much point in actually visiting the shop personally unless you have a lot of spare time on your hand. I wasn't too impressed by Dreamworks when i passed by - there were a couple of frames hidden at the back of the (ridiculously) small store, otherwise they mainly had lots of overpriced vintage European parts - kinda silly going all the way from Europe just to see that. With Punch cycles you can't really guarantee they'll open at 6pm, so maybe have something else to do in the area..
Other bike-ish things not yet mentioned I think in Tokyo: abovebike.com/ , Kinfolk, kuhnsbar.com/entry.html, cycles-yokoo.co.jp/, level-cycle.com/, kalavinka-bikes.com/, tempracycle.com, rew10.com/, sunrise-cycles.com/Home.html
If you've a fixed address somewhere a lot of parts (Nitto, Shimano etc) can be bought on amazon.co.jp at cheaper than store prices - it's easy enough to sign up, they even have an english mode when logged in, and take foreign cards. Any of the independent dealers on the site will take payment and ship in Japan for peanuts (amazon themselves will ship abroad fwiw), far easier than traipsing around the stores to buy the more common stuff IMO.
Eat lots over there, the food is lovely - not just the fishy things but all sorts of fried and curried goods too. Don't forget to drop in to any of the 711/Family Part/Lawson etc stores any hour of the day to stock up on fried chicken from the hot food counter at the front of each store, and other fast food too, it's great fuel for cycling and tourism :)
Also if you plan on catching the Sakura, know that it travels slowly day by day from south-west to north-east (like the autumnal foliage too), you can completely miss it if you're not careful by being in the wrong part of the country on the wrong date, even criss crossing from places it's not really started yet to places where it's almost over. So check your dates.