To get to Juniko, you must first reach the prefectures of Aomori or Akita.
The closest airports are those of Odate-Noshiro (ONJ), Akita (AXT) and Aomori (AOJ).
As of 2018 there are no low cost airlines available for these airports, so your choices are between JAL and ANA.
If you arrive by train, take a Shinkansen (the bullet train) from Tokyo to Shin-Aomori (just over 3 hours) or Akita (over 4 hours), and from there change with the Gono line.
It takes quite a lot of time from Tokyo…
After this, from either Aomori or Akita, change on the “Resort Shirakami” Limited Express train, which will take you along the beautiful Gono coastline until Juniko (the most scenic route is the one coming from Shin-Aomori, from the north). However, keep in mind that this train is quite touristic, and thus very popular at times, so get your reserved seat ticket in advance, if possible.
From Juniko Station to the Lakes Park, a bus is available during summer.
You can check this link below for further information (at the bottom of the page some English basic information is provided)
If you like walking, just know that the station is 5.5 km from the lakes, which could become a pleasant walk!