Hard to give recommendations without knowing you or what you're interested in. Based on others recommendations above, there are a couple of things I'd add.
JapaneseRuleOf7 recommended Mt. Takao. I live just a few stops away from there. Mt. Takao is somewhat known for the statues of Tengu associated with the temples there. It does make for a fun little day trip- it's maybe an hour or so out from Tokyo on the Chuo line (take the commuter rapid if possible, the regular rapid as a 2nd choice, and avoid the local train. It would even be worth it to take the Ome special/commuter rapid- also on the Chuo line- as far as Tachikawa and switch from there.), and then change at Takao station to get out to the mountain . Mt. Takao is nice, but a bit of pacified nature- well trodden paths and it can get a little crowded on the weekends. There's also a Trick Art Museum down by the station if you have a few extra hours to kill, or if you get out there and it's raining.
Quite nearby Mt. Takao is the ruins of Hachioji, though they're a little less convenient to get to- the path up Takao (or the train/chairlift to the top) is close to the station, Hachioji castle would likely require a taxi trip- I biked from my apartment. These are also sort of scattered over a small mountain, but the paths are not quite as commonly traveled and you'll be able to escape the crowds and feel even more "in the forest". It's also considered one of the most haunted spots in Japan, all the defenders (the main force was off to a different battle, so it was under-defended) having been put to death when the castle was taken (and the location lost, not to be re-discovered until the 1980s). Personally, I love ghost stories but don't believe in ghosts, so this was a cool day trip. The anniversary of the fall of the castle is coming up soon- June 23rd, I think.
I might also recommend Tachikawa. Recently it's been connected to several anime/manga, and if you want to spend a day in the park, you can't go wrong with Showa Kinen park, which is very large and has some very cool areas hidden toward the back (the Sun and Moon Pyramids and the dragon playground are very neat). Honestly, though, since you're only visiting Japan for a short trip, you might be better served by hanging out in Yoyogi park and checking out the performers practicing their routines and such.
As for Akihabara- don't miss the 5 floor Kotobukiya store (they carry a lot of other stuff, not just Kotobukiya products). Akihabara restaurants- if you can manage to get in, there's a restaurant called Granvania near Akihabara station (upstairs from the lightbulb shop near the main Akihabara Electric Town street crossing). While technically a maid cafe as the staff wear the outfits, it's more like a regular restaurant in that they have individual tables and don't do floor shows and go overboard with the cutesy crap. The food is ok, not great (but not bad), but I really like the style and the decor of the place.
If you are into figures/collectibles, another place to check out is Nakano Broadway (roughly halfway between Tachikawa and Shinjuku on the Chuo line, maybe 20 min from Shinjuku). Basically it's a big shopping area with lots of stores carrying figures both new and secondhand, as well as a scattering of other collectible shops, cosplay goods stores, 2nd hand clothing, and other assorted shops.
I'm not much of one for the club scene, so I can't really recommend anything there. For restaurants/bars, though, I do like the SF themed bar Kanda Flux (
https://www.facebook.com/kandaflux) over in Kanda (next stop from Tokyo Station). Prices are decent, their food is good, but you might consider it a step above snack food and not quite "meal food" so you may want to eat before going. They have memorabilia from several different SF franchises decorating the place, mostly Western SF stuff, though. There's a life sized Batman, Han Solo in Carbonite, some back to the Future stuff, and the bartender will mix a drink on the spot based on your favorite movie or character.
If you're looking for more of a Japanese traditional restaurant, or Japanese themed, you might want to try Ninja Akasaka (
http://www.ninjaakasaka.com/). I've never been there myself, but have been thinking I ought to get there one of these days.
Cheaper restaurants... Coco Ichi Curry. I've never tried Yoshinoya's curry, so I don't know how it would compare, but I do like the Coco Ichi.
As you're out and about, keep an eye out for a store called Book Off. Basically a large used bookstore, you should be able to find some cheap secondhand manga that catches your interest.
Hmm. You said you're leaving in 3 weeks. No idea if you'd be interested, but on June 7th, if you're around, there's an unofficial Japanese Star Wars Fan Convention, which usually involves a lot of cosplay, fan tables, etc.
https://www.facebook.com/MoistureFarmersUnion
June 12-14 the Tokyo International Toy Show is open to the public.
http://www.world-of-toys.org/japan/tokyo-toy-show/
That's all I've got off the top of my head.