I was there last October, there is a short thread if you search for it.
I highly recommend the book “A geek in Japan”. It covers a lot of places to visit, as well as cultural phenomena.
http://www.kirainet.com/english/a-geek-in-japan-the-book/
I would visit these neighborhoods: Start with Shibuya. Here you have the infamous Shibuya crossing, small stores with lots of youth hanging out, and a giant mall. Harajuku for “kawaii related stuff” and youth culture, Akhihabara for electronics and video game and manga related stuff, and Asakusa for traditional Japanese stuff. In Asakusa there is a street called “kitchen town” (google it) that sells only kitchen equipment - from knifes to Sampuru (the fake food you will see in all restaurants). This is no tourist trap, real restaurant owners come here for quipment. Strangely enough, there are almost no places to eat there. OH! In Akhihabara you can eat at the extremely awkward Maiden Cafes.
You can skip Ginza and Roppongi Hills, it’s the expensive part of town with Prada and Gucci stores etc. Nothing you haven’t already seen in London or NYC or any other metropol. Don’t quite remember Shinjuku…
Go to Tsukiji fish market. You need to be there early, like 5AM for the tuna+bandsaw action. By 10 I think everything is shut down (that’s when we got there). You can get sushi there, quite expensive but well worth it. Near Tsukiji there is an Advertising and Marketing museum which was worth the visit. In the same building (skyscraper) you have fancy restaurants on the top floor. It’s close to the hotel that is featured in “Lost in translation” I think.
For shopping I liked Muji and Loft. Muji stores carry way more stuff than the stuff we have here in Sweden at Åhlens. It might be tempting to go to Muji concept store in Shibuya, but I would skip it - it’s tiny, like 20 sq m - and only has some napkins and crap. I think the biggest one is in Ginza. Tokyu Hands is a store with various fun stuff, kinda like a Japanese version of the Swedish “Clas Olsson”. The one in Harajuku has a Cat cafe on the top floor (a room full of cats that you can pet). I don’t usually care for shopping, but the highlight of my trip was actually looking at things in stores that is not electronics. From bento boxes to brooms, everything is different. The stores are basically a design museum.
We were about to stay 1 night in a Pod/capsule hotel in Shibuya I think, but apparently you aren’t allowed in there if you have tattoos so we had to cancel. Also aren’t allowed to be a female. Cheap Tokyo Hostels » Find 63 hostels in Tokyo » Hostelworld
Watch the Tokyo episode of “an idiot abroad”. I can look at my archive of guide maps and stuff I saved if you’re interested. Definitely going there again. Kinda overdid it last time - wanted to see everything and spent too much time on the subway going back and forth.