My personal advice as someone who basically stays in Tokyo one-third of the year is to stick to Tokyo and its surroundings. The city is so big and there are so many things to see that two weeks go by in a flash, and adding travel time in the middle is just going to be a waste. After eight years of going there, I haven't seen half of it.
First piece of advice right as you get off the plane: if you land in Haneda you're fine, you're already in Tokyo. If you land in Narita, your trip hasn't ended yet. Take the Keisei Skyliner and not the Narita Express unless you plan to stay in Shinjuku, Tokyo Station area, or Shinagawa. It's cheaper, faster, and better in every way. If you want to save money, the Keisei line also offers cheaper but slower alternatives and has comfy stops in Nippori and Ikebukuro.
Incidentally, Nippori station is a good landing zone in Tokyo, as it's smaller and stations like Tokyo and Shinjuku can be VERY confusing for a newbie.
Get yourself a Suica card (or a Pasmo, it's the same thing). You can find it basically at every station. It costs only 500 yen plus whatever you want to charge it with, and it allows you to travel hassle-free for the rest of the trip. The 500 yen is a deposit, and if you plan to never come back, you can get it back by returning the card at the airport before you leave.
Of course, you should probably go to Tokyo Game Show. public days are crowded, hot, and sweaty, but it's one of the best gaming shows out there, if not the best.
In Tokyo, there are just a million places to go. For clubs, you should probably stick to the Shibuya area unless you know what you're doing, because Kabukicho and Roppongi can be pretty hardcore tourist traps. I suggest actually looking up the Meetup app, which has a lot of club going stuff that is a bit more foreigner friendly (but still full of Japanese to socialize with). The weekly pub crawl isn't bad if you can hold your drinks
If you're with them, you're fine in any area, as they'll keep the group out of trouble.
To see some classic Japan, there's plenty you can find around Tokyo. The Takao area is great (Takaosanguchi station, not Takao station) especially if you love Trekking.
The smaller suburban cities around Tokyo can also be a lot of fun for a day trip, including Yokohama, Yokosuka (if you like battleships you must visit the Mikasa), Tachikawa, Hachioji, and if you like a more classic atmosphere with temples and more, you can go all the way to Kamakura and Narita You really don't need to travel far from Tokyo to get into smaller villages and such.
Look up traditional festivals in and around Tokyo. There's plenty in September since it's still summer. Good food, local flavor, and plenty of fun to be had. I believe there's one close to Yoyogi park with plenty of food from various regions of Japan.
Even Shotengai (traditional local shopping streets) are a lot of fun to explore for the local flavor, and there's pretty much one in every neighborhood. If you go to Kichijoji, the Sun Road and Daya Road shotengai are great, and they're going to be a new location in Persona 5 lol. The beef mecnikatsu you can find at Satou in Daya Road is one of the best things you'll eat in Tokyo.
If you're with your significant other, good date spots are Odaiba (a bit more pricey) and Kichijoji. You really can't go wrong with the Odaiba Ferris wheel and a boat ride in the classic pond at the Inokashira park in Kichijoji (i mean, it can be fun even with a friend, but be ready to be surrounded by couples).
If you go to Odaiba, it's fun to take the Tokyo Cruise from Asakusa (which is also another nice place to visit, if a bit tourist trappy, nowadays, but the temple complex is great). Their ships are designed by Leiji Matsumoto, and it isn't too costly (about 1500 yen a person one way, you can do the return trip with the train).
If you're into museums, a good start is Ueno Park, which has plenty, on top of a zoo with pandas.
I never went to sumo fights, so I don't have much experience, but I don't believe they're very expensive since it's a popular sports.
For capsule hotels, there's one in Akihabara that seems pretty fun (only seen from outside). The name is Glansit. I'd say about 1000-1500 $ are fine if you're going to backpack, but that also depends on shopping and eating. Don't go to touristy restaurants, look where the Japanese eat and do the same. It's gonna be yummy and very affordable.
Family Restaurants are fun and affordable, that includes popular chains like Gusto, Saizerya, Bikkuri Donki, Jonathan's, Coco, and Danny's. From the appearance, they seem to be serving western food, but they do it in a really funny "Japanese" way that makes the experience very different from similar places in the west. They also have seasonal menus that offer interesting stuff for cheap. Of course, ramen shops will be your bread and butter, and when you're really hungry, try an all-you-can-eat place for lunch. Eating out at lunch pretty much always saves you money. Don't be afraid to go to chain places. In japan they're much better than in the west. As a matter of fact, it's really hard to find food that isn't good in Japan, even at convenience stores.
I advise staying out of tourist traps. IE: crap like the robot restaurants in Kabukicho. They're pricey and they're ultimately crap. On the other hand, the small bars in Golden Gai (still in Kabukicho) are awesome. Small bars and eateries are awesome period, regardless of where you go. If a place is called something Yokocho, you're pretty much guaranteed to find nice small bars there.
For sightseeing from above, you can go to the Tokyo Government Office in Shinjuku (Tochomae station on the Oedo Line). It's free and great, and you don't need to pay the fees required at Skytree or Tokyo Tower. The difference is really not worth it.
For anything that attracts crowds, try to go on a weekday. EVERYTHING is very crowded in the weekends, and that includes bars and restaurant. Be ready to line up a lot on Friday and Saturday night, even for a seat at an Isakaya. Incidentally, Isakaya are great, but keep a close eye on the bill, because it's easy to go overboard (try to stick to those that have a tablet for ordering, it's often in English and lets you bring up your current bill at any time).
if you have a weekday free, and like Disney, the Disney Sea park is a lot of fun, and a bit more mature-oriented (they even sell alcohol) compared to Disneyland. It's pricey but worth it for a different experience compared even to western Disney parks. do NOT go on a holiday of any sort.
For gaming related fun, of course, there is Akihabara, but I also advise Nakano and its geeky mall Nakano Broadway. The VR Zone in Kabukicho is fun even if a bit pricey. Keep in mind that both Akihabara and Nakano are home to a massively flourishing preowned market, so don't buy stuff new unless you can't avoid it.
As for stores, Super Potato is fun to visit if you like retro stuff but do not buy there. They're basically a tourist trap for gamers and their prices are pretty bad. Surugaya, Trader, and Mandarake are better bets (in that order).
Personally, I don't care for maid cafes, but I know many enjoy them and they're pretty innocuous, albeit a bit pricey. Go if you like, but if it looks seedy, it probably is. If you're into more "adult" entertainment, I'm afraid most don't accept foreigners, and those that do, aren't good, so you're out of luck there.
I just came back from Tokyo Yesterday, and now I miss it already lol. I'll have to wait until September myself to go back.
PS: if you plan to shop a lot, remember to bring plenty of luggage space. It's VERY VERY easy to go overboard on your first trip to Tokyo especially if you're into stuff like games, figures, manga (be careful those are HEAVY) and such: Thanks to the preowned market I mentioned above, they're cheap and tempting.
Finding out that you have to leave stuff behind to get the luggage into the flight is a bad way to end your trip (and shipping stuff back is pricey). If you find yourself over capacity (try to find out at least 3-4 days before your trip ends), buying an additional suitcase is your best bet. You can find some easily and cheaply on Amazon Japan, and they do deliver to hotels quickly.