Regarding visit in June

Please advise what city, towns we can visit Japan in June. We`ve never been before in Japan and I know that June is the most rainiest month. Our flight will be in Tokyo and friend lives in Fukuoko , so we`d prefer to visit another interesting towns located on the way between Tokyo and Fukuoko. Please advise which cities we should visit within 13 days in total? Thanks

Do you mind the heat? June gets very hot so if you dont like heat, look for places in the mountains or go north.

How are your interests and preferences?
This website has various itineraries.
https://www.japan-guide.com/e/e2400.html

"Fukuoko" seems to be a typo for "Fukuoka";
it's a name of both city and prefecture.
Does your friend live in or outside Fukuoka City?

Tokyo and Fukuoka are far away from each other.
On the condition that you leave Japan from Narita [NRT] or Tokyo Haneda [HND],
I suggest that you travel between Tokyo and Fukuoka on one way mainly by air and on the other way by railway.

Weather permitting,
you can take a domestic flight between Tokyo Haneda [HND] and Fukuoka [FUK].
Unlike many other airports in Japan,
these two major airports are located just near urban areas of a mega-city.

ANA and JAL now each set discount prices for passengers from overseas to have domestic flights.
https://www.ana.co.jp/en/us/promotions/share/experience_jp/
http://www.world.jal.co.jp/world/en/japan_explorer_pass/ar/

Dear Lazy Pious, thanks for reply. The arrival flight will be in HND (haneda) airport of Tokyo. Would you please describe me the Itinerary Ideas starting from HND airport to Fukuoka capturing very interesting places within this trip. Duration of stay in Japan is 13 days (12 nights). And what kind of sightseeing we should see in Fukuoka prefectura. Sorry if I made mistakes in spelling of city names and etc. And please consider the low budget (cost saving)

Hi. If I was going that time of year I'd try to stay away from the cities because they tend to be hotter and more expensive. but then I dont like cities anyway! as the above poster mentioned, Fukuoka is a fair way away from Tokyo. What I would do is to travel through the mountains mainly by train - Tokyo to Bessho Onsen - Nagano - Kanazawa - Kyoto (it will be dreadfully hot but you really cant miss it) - Nara - Hiroshima - Fukuoka/Hakata.

Or for a cooler trip, after Nagano got to Matsumoto - Kamikotchi - Takayama - Kyoto.

Have a look at those and see what you think.

This part of the site may be helpful...

https://www.japan-guide.com/e/e2400.html

I also find going north (Tohoku) is great during summer. It's cooler and less affected by rain.