Bratislava's location at the meeting point of three countries makes it an unusually good base for day trips. Vienna is 65 km away and takes an hour by train — a full day in Austria's capital and back for dinner in Bratislava is perfectly manageable. Budapest is 2.5 hours and gets you to one of the great cities of Central Europe. Brno in the Czech Republic is 1.5 hours by bus.
Closer to home: Devín Castle is 12 km west along the Danube, accessible by bike in 30 minutes or by bus in 25. The Small Carpathians wine route starts at Svätý Jur, 15 km north, and runs through a string of cellar towns; Modra, 35 km away, pairs its wine with the blue-and-white folk pottery it has made since the 18th century.
Rusovce Mansion and the Roman camp at Gerulata are 14 km south, and the Danubiana art museum sits on a peninsula in the Danube 30 km downstream. Čachtice, where the "Blood Countess" Elizabeth Báthory lived, is 70 km northeast and makes a longer half-day trip. The Demänovská Ice Cave in the Low Tatras is 3 hours away — a reasonable day trip with an early start.