Any question answered...

I really like de Pijp for mixing with locals (has many of the best bars/restaurants etc), really pretty, lovely vibe and is just far enough out that you're not seeing tourists everywhere and being sold to. But, you can walk/ride/tram into the center in minutes. Van Gogh museum is definitely worth a visit. Personally I found Anne Frank's house a bit disappointing - don't get me wrong, I think the book's amazing - but visiting the house you queue for ages, during which they hand out an info brochure. Then, when you get in, the few exhibits that are there have the exact same info as the brochure. Not really sure what I was hoping for. Maybe some animatronic nazis.

My favourite thing in amsterdam is just walking around, finding different charming streets, having tiny beers in nice little bars, little puff if that's your thing, just leave yourself open to the charm of the city and you'll be hard-pressed not to have a lovely time... if you want any recommendations re restaurants etc let me know!

Soak up the atmosphere with either:

1. a boat trip. One with a bar at sunset is a good shout.
   
2. Take bikes and head south down the Amstel. It's very pretty and you can stop at bars or the beach by the rowing club.
   
   

Eat vlaamse stoverij(?) at the Flemish culture centre. Not a good option for vegetarians.

Just wandering around is a good call, but if you want to wander in a slightly more informed way you can join a walking tour. It's nominally free, although you'll end up tipping your guide a bit. Keep an eye out for various markets, which you can revisit on your own.

The Rijksmuseum is quite impressive and an audioguide can help you do the "greatest hits". I understand the Van Gogh museum is also very good and on my list for next time.

The ferry across the river can take you to NDSM , which is part of a regenerated/ing industrial neighbourhood with lots of artists in residence and you can go to Pilek or Noorderlicht for that sunset beer.