Trains leave hourly for Füssen from the Starnberger Flügelbahnhof, tracks 27-36 on the north side of the Munich Hauptbahnhof. It's about a two hour trip. Sometimes there is one change on the way, usually in Buchloe; sometimes the trip is direct. These are all regional trains, so you can use a Bayern-Ticket (€23 for 1 person, €27 for 2) for the entire trip.
The first train out after 9 AM in the morning, for which the Bayern-Ticket would be valid for the entire trip, leaves Munich Hbf at 9:53 and goes direct to Füssen Bahnhof at 11:54. The bus (12:05) would get you to Hohenschwangau at 12:13, still with enough time to see both castles.
If you want to arrive an hour earlier, you can still use the Bayern-Ticket for most of the trip, but you must buy point-point tickets to the first stop after 9 AM. A train leaves Munich at 8:53 and gets to Geltendorf at 9:21. The fare from Munich to Geltendorf (MVV 3 zone Einzelfahrkarte) is 7,80 per person. Just stay on the same train and change tickets. The train goes on to Buchloe at 9:39, has an 8 minute cross-platform change there, and gets to Fuessen at 10:55. The bus to Hohenschwangau leaves Füssen at 11:05 and gets to Hohenschwangau at 11:13.
The trains arrive at Füssen Bhf, a very small station. For those of you with luggage, there are 30 lockers, 12 small (€2 in 2009), 18 large (€3 in 2009). My carryon size bag fit easily in a small locker. Most were empty when I was there.
There is a bus stop (Haltestelle 2) a few yards from the front door of the Bahnhof. From there buses leave a few minutes after the train's arrival for Hohenschwangau.
View out the front door of Füssen Bahnhof (that bus took me to Hohenschwangau)
Sign for Haltestelle 2 Note castle silhouette
The bus stops in Hohenschwangau. The tourist information office is right next to the bus stop. You can also leave your luggage there.
It's just a 100 yd walk around to the right to the ticket kiosk.
to the ticket kiosk
<--------round the corner
<----- from the TI
At the front entrance to the ticket kiosk, there are two lines. The left-hand line is a serpentine line to multiple counters. It was filled, but seemed to be moving fairly fast. It was for walkup purchasers of tickets that included Neuschwanstein (tickets for that castle alone and for both castles). The other, much shorter line, was for people who already had reservations or for just Hohenschwanstein castle. At the time I arrived, they were booking Neuschwanstein tours for about an hour later.