Getting There
Unless you have the funds to fly, Tohoku is best reached by train. My rule is still to avoid taking the exorbitantly expensive bullet train (shinkansen) whenever possible, so I recommend overnight express trains (the kyuko or more expensive tokkyu). There are a lot of these, and most run either up the Japan Sea coast or further east through Sendai and Morioka. To get to Akita, you'll want the Japan Sea line. To give some examples: the Nihonkai Ichi-go (no. 1) sleeper train leaves Osaka at 5:45 p.m. and arrives Akita at 5:36 a.m. Other overnight trains leave Ueno in Tokyo at around 9:30 p.m. and arrive in Akita at six or seven in the morning. There may also be special kyuko express trains; these are much cheaper but sometimes have no sleeper berths, just cramped seating. Check a current train schedule to see what the offerings are at the time you want to travel.
Accommodations
Some of the youth hostels up in this area are quite tolerable; Tazawa's has friendly staff (and good meals, as I recall). Needless to say, there are minshuku scattered throughout the region as well. In Osorezan, the temples do offer lodgings, but they may fill up with pilgrims in the summer; check with a tourist office, or take your chances; if they're full, cycle back to Mutsu and stay in one of the inns in Mutsu; the people at the tiny Shimokita station were very helpful and can probably assist you in finding something.
Sights
The sights have been pretty well covered in the story itself; needless to say, lakes, greenery and hot springs are the major attractions of northern Tohoku. The long Oirase route is justly famous for beautiful river scenery, though the clogged road running alongside can occasionally take away a bit of the ambience. If possible, try this route in autumn. Onsen (hot springs) are scattered throughout the area, particularly good was Tamagawa, one of the ones mentioned in the text. It shouldn't be hard to plan your route to pass by at least one onsen per day. Note that the Touring Mapple guides note many onsen right on the touring maps.
Getting Away
The nearest train station to the Shimokita area is, quite logically, Shimokita on the main JR line. There is also a private line running between this station and Ohata-cho further up the eastern coast, but it probably won't save you much time, so you won't want to use it unless you're totally exhausted. From Osorezan, you could also take a side road down to Ominato, the end of the JR line, but I don't think this saves much time either. From either station, trains leave every hour or two to Nobeji and most go on to Aomori, where you'll need to go if you're going down the Japan Sea coast.
Three ferries a day travel to Hokkaido from Oma at the top of the Shimokita peninsula at a cost of around a thousand yen. One per day also leaves from Ohata, a bit down the eastern part of the peninsula, bound for Muroran (1400 yen). There are also ferries from Aomori to both Hakkodate and Muroran.
If you're cycling away, I'd suggest avoiding the western shore of the long peninsula leading from Mutsu down toward Aomori, as the single highway seems to be permanently clogged with cars. Also, the western coast of Honshu leading to the Akita area didn't seem all that interesting; you'll probably want to choose another route.