Definitely not worth the time or additional money spent on transit going back and forth when everything you want to see is contained within a fairly compact city. Last May at the beginning and end of a trip to the Netherlands, I stayed at Hotel D'Amsterdam Leidsesquare (airport bus stop about 3 blocks away, hotel is right near Leidseplein, Vondelpark and easily walkable to many of the large museums such as the Rijks and VanGogh) and I also stayed at The Times Hotel (located right on one of the main canals just on the edge of the Jordaan neighborhood and very close to The 9 Streets district and the Anne Frank House). Got decent deals on both hotels - one by using Booking.com and the other reserving direct through their website. You could also consider a private room in a hostel, like Cocomama (which also has 2 bed dorm rooms if you just want something in a good neighborhood that is convenient, cheap and clean).
Hard to suggest on what you should do if you don't specify what your specific interests are. If you are a museum person, there are a ton of them spanning just about anything you could think of (traditional Dutch masters, modern art, photography, Dutch Resistance/WW2, prostitution, purses & handbags, cats, fluorescent "art", historic canal homes, etc.). If you like to shop, there are areas like The Nine Streets with loads of small boutiques and there is also a number of outdoor markets scattered around the city. There are lots of brown cafes if you just want to pull up a chair and have a drink while you people watch. And if you want to see Amsterdam from the water or by bicycle, there are lots of options to do that. Depending on the weather (I would not buy tickets in advance - last May it rained nearly every day and was still pretty cold on a lot of days), there is also a visit to Keukenhof Gardens which is a day trip outside of Amsterdam. The Gardens will be open during your early May visit, but I would not bother with making the trip out there if the weather was not sunny.