Well it is unlikely you could find a more expensive way of doing this Also, Dutch roads, especially in this area (Randstad) are unlikely to offer the possibility of cars pulling over so that you can get out and wander around. Perhaps parking in a cafe, then wandering off might work, but this really doesn't sound like a realistic plan.

I would suggest taking the train to Hillegom, which is a small station between Haarlem and Leiden very close to Keukenhof. From there, you could rent bikes and ride the last 4-5kms (2-3 miles?) to Keukenhof, where they have free bike racks. Or, you could hire a train taxi (treintaxi) which would drive you there for a fixed price (you might have to share it with other people). If the weather is nice, you could even walk.

Leiden is also a very nice old university town. You could take the train from Amsterdam, wander around the town for a while, then take a local train to Hillegom, walk/bike/taxi to Keukenhof, see the gardens, then walk/bike/taxi back to Hillegom, take the train to Haarlem, walk around Haarlem (the central square is quite nice, plus they have nice museums) for a bit, then take the train back to Amsterdam (about 10 minutes). This is probably too much for one day, but you could split your time between Keukenhof and either Haarlem or Leiden.

Have a nice time!