Getting to Slovakia
The easiest way to get to Bratislava by train from west is via Vienna, in Austria. The trip takes 50 - 70 minutes depending on whether the train is an express or a local train . For the north and east of Slovakia, international trains leave from Prague in the Czech Republic to Zilina with some continuing to Kosice.
Trains also travel to Slovakia from Poland, Hungary and the Ukraine. Crossing to and from Ukraine is a lengthy process due to bogie changing (different gauge in Ukraine) and security measures. Direct international trains may be expensive: the best option is to buy a ticket to a station just on the other side of the border, and buy another one for onward travel there. You'll avoid the surcharge that way.
Among many others, there are regular services from Vienna, Prague and Budapest to Bratislava; and from Uzhhorod, Ukraine to the eastern Slovak town of Michalovce and from Krakow, Poland through Zakopane, Poland to Poprad.
From Budapest the travel is 4 hours, the bus stop for 5 minutes at Gyor and in a small restaurant in the road.
Bratislava has its own airport. Sky Europe was the main airlines but to illiquidity has suspended all flights for the time being. RyanAir operates to Stansted, Hahn and some other cities.
Full service carriers providing service to Bratislava BTS are Czech Airlines and Lufthansa. Czech Airlines (CSA) provides several flights a day to/from its Prague hub. Czech Airlines connects its Praha hub also with Slovak airports in Kosice, Sliac and Zilina. One can also fly between Kosice and Vienna (OS) and between Kosice and Bratislava (NE). Similarly Lufthansa (LH) provides several flights a day to/from its Munich hub. There is also nonstop connection to Moscow and several other cities to the east.
The other alternative is Vienna airport Schwechat, which is just about 35 kms from Bratislava. It provides a more convenient way of arriving to Slovakia by the major airlines, but can be more expensive. Buses leave for Bratislava hourly, optionally you can take the airport shuttle.
Poprad - Tatry Airport is also connected with Bratislava, Bologna and Basel thanks to scheduled routes operated by Danube Wings.
You can also fly to Krakow if you want to go to the Tatra Mountains. Buses from Krakow run to several Slovak towns around the Tatra mountains and Orava.