The easiest way to get out to the Keukenhof Gardens from Amsterdam on your own is to take the direct bus from Amsterdam Airport Schiphol. The Connexxion bus organization (a national transportation system, not a tour company) offers a "Combi Ticket" from Schiphol that includes a 35- to 40-minute bus ride plus the entrance fee to the Keukenhof, which totals € 19.
Buy tickets from the Amsterdam Tourist Information Offices (called "VVV" offices) for Bus 58 from Schiphol direct to the gardens.
The buses leave every 15 minutes from the second island outside the arrivals-level entrance at the airport. See the Keukenhof website for more information.
Note: there are tour outfits that charter buses for groups to go to the Keukenhof. Ask the VVV offices for tour group information.