You can find most answers to your transportation questions by referring to the japan-guide.com pages for your intended destinations, and by plugging in station names on route search engines like Jorudan and rome2rio.com. If you've been living in Japan for a year, you'll have gained enough familiarity to piece the routes together.
If you don't want to spend much money, then I'd advise you to pick either Hida-Takayama/Shirakawa-go or Hakuba/Nagoya, unless you want to whittle away large chunks of your time on buses and slow trains while dealing with inconvenient/impractical connections.
It takes 2.5 to 3 hours to reach Hida-Takayama from Tokyo by shinkansen, 8+ hours by highway bus. No matter what route you take, you'll probably have to take another bus out to Shirakawa-go. When I visited those places a few years ago, three days were barely enough for round-trip transportation and selective sight-seeing... and I traveled by the fastest means possible.
By "climbing" do you mean rock climbing or hiking/trekking? I was an avid four-season mountaineer in Japan, and I've never heard of any notable rock climbing areas in Takayama/Shirakawa-go. Even the mountains immediately surrounding the area are rather low and grassy. Haku-san (Mt. Haku, ~2,700m) is a lovely mountain in that general area, but further out and can be a bit of a pain to get to, depending on the season. If you really want to hike up a more alpine-like mountain in Japan, and don't want to spend money on shinkansen, then Haku-san may be a more practical alternative to going all the way down to Hakuba. Incidentally, Haku-san is one of Japan's Three Holy Mountains, so it's not some obscure, backwater mountain. It also has several peaks, so you can chain them together into a multi-day hike. Since there are unmanned emergency huts with no electricity/water and a large manned hut with all conveniences, you wouldn't need to worry about carrying a tent, if you plan properly.
The Shirouma range in Hakuba is stunning, especially in autumn. I'm assuming you want to go up the Daisekkei? I realize it's talked up a lot in guide books and by Japanese hikers, but it's not particularly special, and can be a bit dingy late in the season. Just make sure you take a pair of mini crampons, at the very least, in case the slope is particularly slippery, or if you're not confident on snow. Although I climbed it easily without any crampons, I had good shoes and the snow conditions were perfect. Most people wear intermediate crampons. Full crampons are overkill in August.
Shirouma also has numerous tall peaks you can chain together, but I'd advise you to stay on the main peak of Shirouma-dake, unless you're an advanced hiker with solid endurance and scrambling experience, and possess both a good map and route navigation skills. There are some ridges that are extremely exposed and treacherous, even though they don't require a high technical ability.