Last minute trip to Amsterdam- car?

Although not mandatory, a car will make your trip much more fun.

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We stayed in Leiden last year at the Huys van Leyden (http://www.huysvanleyden.nl/eng/home-english.html). Parking was not free, but we were able to park for a fee in one of their locked garage spaces in the back. We walked everywhere in town and thoroughly enjoyed what we saw there. We especially liked the Molen de Valk Windmill Museum (http://molendevalk.leiden.nl/en/).
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We loved driving all around the area and seeing the fields. Driving was easy. The Keukenhof Gardens (http://www.keukenhof.nl/en/) were spectacular, even though it had been a very cold winter and the tulips weren't strongly in evidence. I had been before and never seen the greenhouses, so seeing them was a real treat. Parking was not free there either, but it was easy, largely because we went there on a Monday instead of the weekend.
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One thing we missed was the Den Hartogh Ford Museum (http://www.fordmuseum.nl/cms/pages/english.php?lang=EN) in Lisse. We just ran out of time.
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A fun surprise was running into a bunch of vintage American cars on the roads. There seemed to be a preference for Studebakers and Cadillacs. My husband insisted that I document driving to and from Keukenhof, so if you are interested PM me and I will send you the link to my blog entry about that.
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Haarlem is within easy striking distance of Leiden and Lisse (Keukenhof) and Amsterdam. We visited the Franz Hals Museum (http://www.franshalsmuseum.nl/en/) and Corrie Ten Boom's House (http://www.corrietenboom.com/history.htm) there.

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Also within easy driving distance are the beach resort towns that you might want to explore. By "easy" I mean within 30 minutes depending on which route(s) you take. Google map the area and you will see what I mean.
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Likewise, taking the train from most of the locations in the area will take about 30 minutes or so, but we stayed in Amsterdam for that part of our trip. We liked being able to walk around or hop on a bus or tram to go everywhere.

Travelling by car is easy. Just be aware of speed limits, including the principle of 50km/h limit withing city/town limits. Highways are immaculately well kept and a breeze to drive - the only major recommendations are beware of speed limits and don't hog the left lanes unless overtaking.

A car will allow you to explore the flower fields outside the park. You can also visit the dunes, or take a side trip to the big levees and lakes in the North, where there is a second flower-growing area full of color this time of the year.

Most Dutch car rental agencies don't charge fees for returning the car elsewhere, and when they do this fee is low (€ 15-30 maximum).

Since you'll have a car, why don't you stay in a hotel near the dunes, or by the coast, and then go to places you want to visit? Several secondary train stations have Park + Ride facilities, so that if you want to visit big cities you can comfortable travel in and out. Utrecht and Amsterdam also have local Park + Ride networks near some freeway exits (they are signed as P+R).

What the others said... definitely get a car if you have a love of photography. It will allow you to be at the right place at the right time and take as long as you need to get the image you want:) It has been a few years since I drove there, and without a GPS and really not a good map I did get lost a few times in Amsterdam trying to get out of the city, but we eventually found our way.
Last minute travel is a really fun way to roll... You have lower expectations, so everything is a surprise!

Just north of Amsterdam you can explore with the car the country-side. Little places like Durgerdam, Waterland Ransdorp, Broek in Waterland, Monnickendam, Edam and Marken etc. are absolutely lovely and worth a visit. You can combine it with a visit to Zaanse Schans.
The road along the Vecht down-stream from Utrecht is well-known for it’s 17th century mansions. West of Utrecht castle De Haar with tiny village Haarzuilens.
The country-side south-west of Utrecht with places like Oudewater, Schoonhoven and (river-ferry to) Nieuwpoort on the way to the windmills of Kinderdijk is a nice route to take.
Some 70km east of Utrecht is Palace Het Loo and nearby Kröller-Müller museum with it’s Van Gogh collection to recommend. Further east the cities along the river IJssel, like Zutphen, Deventer, Zwolle and Kampen (nearby Elburg).
Another few rules, avoid traffic-jams on highways during rush-hour and there are not only speedlimits of 50km/h but also of 30km/h in urban area’s to be aware of.

Thank you to all contributors, your suggestions will be most helpful.