Well… that depends quite a bit on who you are as a person and what you enjoy in your day to day life.
Ultimately, if you are in Tokyo for as long as three weeks then you can do quite a bit. If you like anime and electronics (ESPECIALLY if you like both those things), then you must visit Akihabara (THE largest electronics district on earth)… there are some verrrrrrry cool stores nestled underneath the buildings that are havens for the technically inclined… plus… anime everywhere.
Asakusa is pretty neat from a historical perspective. Tokyo Tower and the huge temple very close to it are pretty neat.
There is also Odaiba and Palette Town (where you will also find the Toyota Showroom and some sweeeeet F1 Racing Simulators, plus the largest Ferris Wheel in the world is… get on the Crystal Box line (all see-through)!
Go to Omotesando Hills / Harajyuku. That is a pretty nifty area, and Yoyogi Park, as well as the Meiji Jingu.
Check out the Shibuya Scramble at the Hachiko Exit - that’s worth looking at just for the sheer volume of people.
Shinjyuku Park costs about 200-300 yen to go in, but it is a truly amazing park, and worth every yen.
If you go to Ginza, that is one of the faster ways to get yourself to the castle in the middle of the city.
If you want to get out of the city a little bit, hop on a train (the Keheintohoku line, for example) and head out to the Saitama Shintoshin. Or in the other direction to Yokohama. Plus, there are a huge number of parks and nooks and crannies around Tokyo city that are pretty amazing. Such as if you go to Ueno, from the west exit you’ll find the Ueno Cultural Centre (Ueno Bunka Kaikan) with an outdoor museum and Ueno park (which has lots of neat stuff to discover). Or if you head out the lower south exit, there’s a fantastic bakery inside the station, but just outside is a bustling vendor street (very busy, foot traffic only), where, under the Tokyo railway line, you’ll find some incredible local ramen shops.
Roppoingi… well, it has some amazing movie theatres, and much of it is pretty classy in the daytime. But it is a place where ex-pats (expatriates) go to party… it is a party district and, unless that’s your thing, it isn’t really a whole lot of fun. I went there only a handful of times when I last lived in Tokyo… for television auditions, to see movies, and once for a small birthday party.
Oh! Ikebukero! Definitely an interesting part of Tokyo to visit. Sunshine Plaza has (or at least had) a massive Ultraman Statue! Plus, there used to be (might still be) this really amazing icecream parlour called “Milky Way” that had giant parfaits that were sooooo tasty!
As for other cities in the future: Sapporo, Hakodate, Kushiro, Chiba, Osaka, Kobe, Yokohama, Chima, Tochigi, Arashiyama, Shizuoka, Takamatsu (and visit Naoshima if you like art galleries), Fukuoka, and of course, Okinawa. And really, any number of places! There are so many incredible things to find in this country! I’ve been here a goodly while (a few years) and I still love living here.
Have fun in your adventures! Be safe, and remember that cities are the same anywhere in the world… even though Japan is a LOT safer than many other places.
As an aside, a very close friend of mine is a Physics professor at Nicolaus Copernicus University in Poland, and she said that she has never felt safer anywhere in the world than she does in Japan. She absolutely loved it here while she stayed.
I fully agree with what that elderly man said… get out of the box and explore. Chase your dreams and passions. Also, if you want to get into some cool programming… check out the Super Data Science programs by Kirill Eremenko and his counterpart (Deep Learning, Machine Learning, and AI Programming) - they’re really first rate courses. I’m studying them now, and they’re super helpful! Also, as a person who has always chased his dreams… it’s really hard, and many many many people will tell you to give up and just do something easy. But… resist… don’t give up. Follow your dreams and passions. It is absolutely the most worthwhile thing in the world when you do succeed (but remember, for every 10 dreams or goals you try to achieve, you will make just one sort of work… of every 10 that just sort of work, ONE will come true… so… never give up trying new things!! “It is better to have tried and failed than to never have tried at all.”)
Best of luck in your adventures. And if you are in Tokyo for the Japanese long weekend (the 15th, 16th, and 17th of July), you can go to Tokyo Summer Comiket!! The LARGEST anime convention on earth! Wahooooo!! (At Tokyo Big Site in Odaiba!)