Feldkirch, Austria and Lake Constance make for wonderful destinations.
It is entirely possible to encounter the royal family at the Kunstmuseum, coming in and out of Schloss Vaduz or skiing during winter time. This is one benefit of such a small country. They are recognizable in their cars, which use their birth year for their licence plate number.
There are a few hotels in Liechtenstein, but they tend to run on the expensive side. There is one youth hostel located in Schaan, but it closes for the winter. You will probably be able to find cheaper accommodation in neighbouring Feldkirch, Austria.
Liechtenstein has a continental climate featuring cold, cloudy winters with frequent snow or rain, making the country a moderately popular ski destination. Summers are cool to moderately warm, also often cloudy and humid.
Schloss Vaduz - This imposing and historic castle, home to the royal family, overlooks the city of Vaduz and can be seen closer by following the Castle Trail starting in Vaduz. The castle is not open to the public, but it is possible to view it from quite close up.
Fast trains from Zurich and many international train services serve Sargans during the day. Consult the SBB timetable  to find out what'll be quickest when you're travelling. There are lockers at Sargans station so you can leave your luggage there. The link  provides a plan of the station showing the lockers and where to catch the bus from.