Your final destination will be Udine train station. Udine has only one train station. The official hotel is less than four minutes by bus from the train station (line 4 "San Gottardo", stopping at Piazzale Oberdan).
For train timetables and other information we recommend the following sites:
If you are traveling on a day in which you badly want air conditioning, consider that most first class coaches have it, whereas on some trains second class coaches do not. You may want to ask when buying the ticket.
During the day trains travel between Udine and Trieste (East of Udine, takes about one hour and 10 minutes, slightly longer with some trains) and between Udine and Venice (South of Udine, takes about 90 minutes with a good train) at least every hour. Most trains between Udine and Trieste also stop in the port city of Monfalcone.
An independent private rail company connects Udine with the town of Cividale about every hour (takes less than 20 minutes). Although trains to Tarvisio (a 50 to 80 minute journey depending on the train) are quite frequent, only two or three trains a day have the most convenient connections to Germany and Vienna (Udine-Vienna takes about 6 hours and 20 minutes, with the right trains).
Within Italy you can build most connections by changing at Venezia Mestre (or Vicenza, or Cervignano, in some very rare cases which even the online systems have difficulties building automatically).
If you wish to manually build trips consider that trains from Northern Europe reach Udine either through Salzburg-Villach or through Innsbruck-Verona-Venezia Mestre. Munich-Udine by train takes less than 6 hours (changing in Salzburg and Villach, or you can take a different train with a single change in Villach, but it takes half an hour longer). From the East they go through Trieste. If you plan to travel from France there are good connections through Turin or Geneva-Milan, however airfares to and from France tend to be competitive with trains.
Please note that Venice (which is an island city connected to the mainland via a long road and rail bridge) has two train stations: one is Venezia Mestre (also referred to as Mestre, which is the port city facing Venice island), and the other is Venezia Santa Lucia (also abbreviated as Venezia S. L., which is on Venice island). All trains to and from Venezia S. L. have to pass through Venezia Mestre (almost all stop there), which makes Venezia Mestre the best destination for connections towards the south and west (e.g. to Verona, Milan, Florence, Rome, etc.), and Venezia S. L. the best station to visit Venice. But if you want to visit Venice your ultimate destination, with or without changing in Venezia Mestre, will be Venezia S. L.
The small mountain resort of Tarvisio, being a former border post and due to a change to higher speed rail tracks which go a slightly different route compared to the previous tracks, has four different train stations/stops (which makes people in Udine, who have their own language, but only one train station, quite envious ;-). The new station, which is the one you should normally refer to, is called Tarvisio Boscoverde.
If you board a train without a ticket you must actively and promptly go to the train crew to notify them that you need to purchase a ticket (you will pay a surcharge).
Before boarding a train you must stamp your ticket in one of the many yellow ticket stamping machines. If the ticket machines are broken you are supposed to take a pen and write the date and time of boarding on the back of the ticket. This is meant to prevent "recycling" of unchecked tickets, but there is an unwritten rule saying that foreigners will not be fined if they don't stamp the ticket (apparently only some Italians have a tendency to feel stupid if they don't always try to cheat the system). The yellow machines are also fashionable to date+location-stamp anything, e.g. postcards, photographs, etc. Usual disclaimer: if you stamp non-ticket items and test unwritten rules you do so at your own risk...