Amtrak doesn't go to Long Island, so it's not on their website. Long Island Railroad is owned by Metropolitan Transport Authority, so it's on their website.
At New York Penn Station, you have to change to Long Island Railroad's Huntington train (timestable is Pt. Jefferson train), however Westbury has LESS train service.
I recommend once you get to NYC Penn Station, walk over to 34th St-Herald Sq on 6th Av-Broadway and take Jamaica-bound (F) train to 169th or 179th Sts for NICE Bus n22 Hicksville. At 169th St (F) train, n22 picks up at 170th St and 182nd St side for 179th St (F) train. During weekday rush hour, n22X bus gets you there quicker.
n22/X has more service to Westbury than LIRR's Huntington train. n22 operates along Prospect Av in Westbury.
You could also take n24 bus along Old Country Rd in Westbury, however during weekday hours of 6am and 7pm, you have to change at Roosevelt Field Mall for connecting n24 bus.