score:4
From Schiphol airport, you can take a train to Amsterdam Centraal. From there you can also get a direct connection to Leiden Centraal.
You can use the Nationale Spoorwegen (National Railways) website to plan your trip accordingly. They offer an English version of the site.
If you are also looking for additional public transportation time tables, for example to use trams or buses, you can use the 9292 Openbaar Vervoer (Public Transportation) website. They also offer an English version.
The best thing about 9292 is that it allows you to plan trips with multiple types of transportation, meaning you can simply enter your starting point and destination and it will present you with all public transportation timetables, including trains, trams, buses, etc.
You can go as far as simply entering addresses and the planner will tell you how many minutes to walk to a bus stop, how long on the bus, how long walking to the train station, when to transfer, walking to a tram station, and finally walking to your destination address.
Both sites also display updated delays.
Upvote:5
The city center is indeed near Amsterdam Centraal. There are multiple trains going there from the airport. It is about a 15 minute train ride. The right platform is signed quite clearly. Be careful to not take the train to Amsterdam RAI or Amsterdam Bijlmer ArenA, these are not the town center.
To go from Amsterdam to Leiden, you can take two different routes. Both will show Leiden CS on the sign and take equally long, so this won't matter. One goes through Haarlem, one through the airport.
You can plan your route at any time on 9292.nl. The trains you need go all day, although less frequently during nighttime.
You need a OV-chipkaart to take public transportation in the Netherlands. You can buy a single-use one at every train station, but these are not usable in other public transportation. It may be worth investing in a card of your own. Check the OV-chipkaart website to find out which card is best for you.