I've been four times now and have gone far south as Kumamoto and as far north as Ishinomaki which is about 60km north of Sendai. I'm also planning another trip for summer 2018 (very vague plans) to see Wonder Festival Summer 2018 and Comiket 94, and am planning to see as many obon, tanabata, toro nagashi and fireworks festivals between those two fan events.
For mountainous areas, you may want to go out to go out to Hakone (includes hot springs) or even out to the Fuji Five Lakes district or maybe Nikko (also has hot springs). If you want to go on the shinkansen (bullet train), you can get to Hakone by shinkansen; from either Tokyo or Shinagawa station to Odawara station, then on to from there Hakone-Yumoto Station via the Hakone Tozan Line.
For the Studo Ghibli Museum, you will have to prebook tickets before you leave via JTB or whatever the equivalent is in the UK. If you can't get in to see the museum (and it's REALLY hard to get into the gift shop in the museum anyway, it's always packed), there are a number of
Donguri Republic stores dotted across the city. Big retailers like Bic Camera, Yodobashi Camera and all department stores will stock a range of Ghibli merchandise anyway.
A few warnings about Tokyo summers (which has me umming and ahhing about my trip); it's the school holidays so it's going to be really busy, it's going to hot and humid as hell, travel might be a bit of problem especially on the shinkansen (travel early in the morning and late at night for the best chance to get tickets) and typhoons and intense summer downpours can occur.
Also a few things that on line guides on Japan just don't seem to talk about; for the love of god always keep you passport on you; cops can check you anytime they like and ask you for identification, no passport on you and they can hold you for up to 23 days or someone from your group or hotel brings your passport to the koban or station they're holding you at. However is is highly unlikely you'd be stopped by a cop. Also JR Rail Passes are a complete rip off unless you're using shinkansen every second day or so. Do some calculations on
Hyperdia and compare the costs with whatever JTB or your travel agent is selling the Rail Passes for. You may want to get an
IC card like a Suica card which covers JR, subway and all private railways in metropolitan Tokyo, unlike the rail passes which only cover JR trains. Convenience stores are fantastic for cheap meals (bentos usually cost less than 500 yen), however the rapid fire questions the clerks will throw at you are confusing.
Learn up on what the heck they're asking you. In fact it might be good to learn some basic Japanese before you go as very few people know any (decent) English. Also while more and more places now accept debit and credit cards, cash is still king, so get some yen before you go. You can get withdraw cash from some ATMs inside Lawsons and 7-Eleven convenience stores.
That's pretty much all I can think of now. If you're stuck on where to eat, I use
Tabelog or
Gurunavi. I fell in love with the place when I went about five years ago and always look forward to coming back.
