1. Walking Tour
This free(ish - you are expected to tip) walking tour is a great idea for your first day in the city. It will give you a good overview of the city layout, along with some really interesting history. It does go through the Red Light District, but the tour guide I had was tasteful, and for me it was a lot easier to go through with a group then just on my own. Plus then you get it out of the way (because you're obviously going to go at least once).
2. Van Gogh Museum & Museumplein (Museum Square)Of all the art galleries I've been to in Amsterdam, this is by far my favourite. I loooved it. It had loads of information along with all the art, and they have amazing tours as well. It's also a really good size, not too small that you feel like it was pointless, but not so big that you feel overwhelmed. It is a bit pricy at €15, but I really did love it! And conveniently it is located right in Museum Square (or "Museumplein" in Dutch). This is the square where you can find the now famous IAmsterdam Sign, along with a few other museums and art galleries and tourist attractions. Plus if the weather is nice, there are a few cafe's where you can sit and enjoy a nice cup of tea or a cold beer.
Side note: I'm going to go out on a controversial limb and say, don't bother with the Rijksmuseum. Unless of course you're really into art, and will actually appreciate the classics. I found it to be far too traditional for my taste and WAY to crowded to be fun, also, it's soooo massive that it kind of just became over whelming for me.
3. Anne Frank Museum*
*From talking with others who haven't read her book and went to the museum, it seems it is slightly more meaningful if you have previously read her book (which I also highly recommend!)
I have already been here twice, and I think The Anne Frank Museum is the most beautiful and special museum I've ever been too. Having read the book only a month before I visited the first time, it really made the book come alive. It felt like I was getting to know her, Anne, as a real person. It has a special feeling to it that I can't really describe, and I just think it's beautiful that this museum exists.
4. Wander along the Prinsengracht canal (or any canal really) and into the Jordaan area
You'll be surrounded by canals even if you don't try, but I found it nice to take some time to really enjoy the view. Take some time to walk around the outermost of the main canals (The Prinsengracht) and take a stop off in the cute and hip area around the Jordaan (lots of shops and places to stop for a beer or coffee).
I definitely plan to add to this, or make a part 2 after I've seen even more things in the wild city, but for now, I hope this gives you some ideas, and keeps you busy! :) Have fun!