Been there 2 times. First time I city hopped a bunch of places over 10 days (Tokyo, Kyoto, Hiroshima, Kanazawa). Second time I stayed in Osaka and just visited nearby cities (Nara, Kyoto).
- Recommend a JR pass as others have said, especially if going long distances.
- Hire a pocket wifi. They cost is (or was) between £40-60 depending on your plan. Very useful for keeping on top of Google Maps and itinerary etc, as well as keeping occupied on the train journeys.
- Hyperdia app is very useful. It pretty much suggests the exact train/line you should take to reach whichever destination you want, including times and such. Great if you're doing a lot of exploring.
- Budget-wise, obviously it depends on your eating habits and whether or not you like to dine a bit fancier, but I mostly just got quick, hearty meals in and spent the rest of the time exploring. With that in mind, I would budget for about ¥3000 (minimum) for your daily meals, then another ¥500-1000 for drinks (it's hot! Use the many many vending machines and stay hydrated!) as a bare minimum. Then your entry fees and travel fees and souvenirs etc. on top of that. This is just for 1 person. I would spend anywhere between ¥3000 on a good day, to ¥8000-9000 including a few goodies or presents and splurging a bit on sushi etc.
Kyoto is quite quaint and "traditional" feeling. I liked it a lot. There's Arashiyama/Bamboo Forest here, as well as other tourist places like the Golden Temple. Nara is nearby, where deer roam about freely everywhere, so that's a bit of a quirky half-day trip.
I enjoyed Hiroshima. It seemed a bit more Westernised and more English-friendly compared to the other cities (even Tokyo/Osaka). The memorial/museum is a must-see. There's also a little island nearby called Miyajima which you can reach by a ~40min ferry trip. I went there thinking it would be a nice quiet afternoon to wander about, but there happened to be some festival going on that exact day and it was absolutely chock-a-block with young couples wearing yukata/jinbeis and street food and a big fireworks show. It was quite the thing to stumble into!
Kanazawa I also enjoyed. It's a bit out of the way (about 3hrs away from Tokyo), but again it's a bit more of a traditional feeling city, somewhat similar to Kyoto.
Reminiscing now makes me want to go back