We recently got back from a visit to Tokyo (4th or so visit) and we stayed in Shibuya at an APA. I think you guys would like staying in Shibuya as we found it lively both during the day and night. Even though we didn't book through them, I'd suggest looking into AirBNB as there seemed to be some good options when we researched it. I wouldn't rule out Shinjuku as it isn't too far away and IMO seems to have more cheaper options.
The suica card is good in that it give you access to all the key transportation options. However, I'll also suggest the 24hr/48hr/72hr Tokyo subway ticket available for visitors for unlimited travel only on the subway lines. The 72hr card at 1500yen is good value IMO. If I'm not mistaken, you can buy multiple cards to cover the length of your stay as the countdown starts when you first use the card. The main disadvantage IMO, is that you don't get access to the Yamanote JR train circle line which can be more convenient/direct for some trips than needing to do transfers in the subway.
To get your feet wet for drinks, I'd suggest checking out the Hub chain of pubs. They're expat friendly with a mix of locals.