To get to Akita from the Tokyo metropolitan area, you can either fly or use the Shinkansen.
There are two airports in Akita: Aki Airport (nearest to Akita City, Yokote, and Tazawako) and Odate-Noshiro Airport.
The Odate-Noshiro Airport is the most convenient for the Noshiro area, but there are only a few flights to Odate-Noshiro Airport from Haneda by ANA.
The flight from Haneda to Odate-Noshiro Airport takes about one hour.
From the airport to Odate City, take the limousine bus (980 yen one way for a fee), which takes about one hour.
From the airport to Noshiro City, you can take a shared cab for 1,200 yen.
It also takes about one hour.
To get to Akita Station by Shinkansen, take the Tohoku/Hokkaido Shinkansen (bullet train), which takes about 4 hours from Tokyo Station.
The Odate/Noshiro area is a transfer from there.
It takes about 50 minutes by limited express train (Tsuruga) and about an hour by regular train.
If you take the train from Akita to Higashi-Noshiro Station, there is a train every hour or so, but the connection to Noshiro Station is not good.
For this reason, renting a car at Akita Station would be a more practical choice.
Here is a guide on how to get to the Odate and Noshiro areas and how to get around the city.