If you're going out for food the best place I can recommend is Shin Hi no Moto (a.k.a. Andy's Izakaya). It's run by a friendly English guy (Andy) and his family. It gets really busy later on so you're best to phone and book on +81 3 3214 8021. It's okay, you can book in English. They do amazing sashimi platters and big mugs of beer. Just make sure you don't order rice (there's no rice in an izakaya and they might scowl at you if you do). You can get there by taking the Yamanote line to Yurakucho.
For general daily eats, I'm addicted to Yoshinoya. Especially their Gyuu-don. Tasty, healthy and cheap. You can find Yoshinoya everywhere.
West Central Tokyo
If you're going to be there over a Sunday, you have to go to Harajuku. Even if you aren't there on a Sunday, the walk up Takeshita-dori is great fun. Here's a map of a little walk you can take up Takeshita-dori, round Harajuku and down to Shibuya:
North-East Central Tokyo
If you fancy some culture, try this route. It takes in the Imperial Palace, Sumo museum and Edo-Toyko museum. You can finish off in Akihabara for sheer geek awesome or save that for another day.
Some people called me crazy for enjoying it but I like the walk across the rainbow bridge to Odaiba. Take the Yamanote to Tamachi and wander east-ish. You'll see the bridge once you're closer to the shore. You can take the lift up to the start of the walk and then wander out for some amazing views. It is, unfortunately, very noisy due to all the traffic but it's worth it. Head right across the bridge and follow it down, it'll probably take about an hour. Once you're on dry land again on Odaiba, you can wander around the shopping malls there (Aquacity, Seaside mall), take in the Statue of Liberty, go for a bite to eat and eventually head back. If you time it for getting dark, you can either get some amazing views of the bridge lit up at night or just enjoy the Yurikamome ride back (it's a completely automated train with no driver).
For the best view across Tokyo, the Tokyo Metropolitan Government building (a.k.a. TMG or Tocho) really can't be beaten (especially as it's free to go up). It's in Shinjuku.