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Austria is a member of the Schengen Agreement.
There are no border controls between countries that have signed and implemented this treaty - the European Union (except Bulgaria, Croatia, Cyprus, Ireland, Romania and the United Kingdom), Iceland, Liechtenstein, Norway and Switzerland. Likewise, a visa granted for any Schengen member is valid in all other countries that have signed and implemented the treaty. But be careful: not all EU members have signed the Schengen treaty, and not all Schengen members are part of the European Union. This means that there may be spot customs checks but no immigration checks (travelling within Schengen but to/from a non-EU country) or you may have to clear immigration but not customs (travelling within the EU but to/from a non-Schengen country).
Please see the article Travel in the Schengen Zone for more information about how the scheme works and what entry requirements are.
To stay longer than 90 days, a non-EU foreigner will need either a long-stay visa (valid for up to 6 months) or an Aufenthaltstitel (residence permit). Visa-exempt visitors may be able to acquire a residence permit inside Austria after entry, but consulates do not recommend this route due to processing times for the permits and that the permit must be obtained within the initial 90-day period of stay.

・EU, EEA and Swiss citizens, as well as non-EU citizens who are visa-exempt (e.g. New Zealanders and Australians), need only produce a passport which is valid for the entirety of their stay in Austria.
・Other nationals who are required to have a visa (e.g. South Africans), however, must produce a passport which has at least 3 months' validity beyond their period of stay in Austria.
・More information about the minimum validity of travel documents is available at the Austrian Foreign Ministry's website.

There are 6 airports in Austria with scheduled flights. The most important international airport is Vienna which has connection to all major airports of the world. Other international airports include Graz, Innsbruck, Klagenfurt, Linz, and Salzburg which provide domestic flights as well as connections to some European countries. Those airports are particularly popular with cheap airlines such as Ryanair. For traveling to the western states it is recommended to use the very close Munich airport.
The most common airports to visit Vorarlberg are Altenrhein (Austrian), Friedrichshafen (Ryanair, Intersky) and Zurich (Swiss).
If visiting Austria for winter sports, choose airport considering cost and duration for the whole trip (plane+transfer), not always Vienna and even likely not in Austria.
Unlike many countries, getting in to Austria for skiing shouldn't imply flying to the capital city first. Vienna itself is a 4 hour drive away from the nearest medium-sized resort, and longer by public transport. See more in GetIn section of Winter sports in Austria.

Bus is not always the cheapest way to travel though deep discounts for advanced bookings are being introduced for long-distance travel (as far as from Warsaw to Austria for €4). The bus may also be the cheapest option if you want to travel at short notice or if you have large amounts of luggage. Bus travel is especially interesting for those comming from the East as there are many buses into Vienna and they are often faster than trains. Information about their assorted services and pricing is can be found in that section.
Eurolines Austria is the largest operator and organizer of bus travel in Austria though many services are not included in their scheduals.

Austria has plenty of connections with all its neighbors daily. Every neighboring country (even Lichtenstein) have trains at least hourly. Many (Czech Republic, Hungary, Germany, Slovakia, Switzerland) even more frequently. The ÖBB (Austrian Railways) operate high-speed ICE and RailJet trains from cities like Zurich, Munich,Frankfurt, Passau, and Budapest. Eurocity trains are the next fastest trains available as well as the trains connecting the bigger Austrian cities called Intercity. Regional trains called EURegio and simply Regionalzug are also avialible from all 8 of Austria's neighbors.
Vienna is a the largest railroad hub but day and night trains from most Central European countries travel to many stops across Austria. Day trains are normally much quicker than night trains. Tickets can be purchased from certain locations to Austria via the ÖBB website[4]. Always compare fares from the departure or even transit countries' railways as there may be price difference even for the same train. ÖBB offers discount 'SparSchiene' tickets to and from destinations like Croatia, Denmark, Germany, Italy, Netherlands, Poland, Serbia, and Switzerland for a flat-rate (i.e. €29 for a one-way seater, €39 for a couchette, or €59 for a sleeper). There are a limited number of tickets at this price. At peak times you need to book in advance. Additional offers are avaible to all countries in Central Europe, although many cannot be booked online.