official site for visitors to Bratislava  

For mountain-bikers, the forested hills – the Small Carpathians – north of the city are filled with trails.  And even if you don't have a mountain bike, there is a network of sealed walking and cycling paths that continues through the woods to historical towns on either side of the range, such as Stupava in the west and Sväty Júr in the east.  One of the best places to access the trails is via the Železná Studienka valley, a short way out of the centre.  Trains go there from the main station if you don't fancy navigating the suburbs. If you plan to take to the trails it's worth buying a map (widely available at bookshops or stores like Tesco) as it's easy to get lost. A good place to aim for is Biely Križ, a meadow high in the forest (it's marked on most maps) where you can relax and enjoy a drink or snack from the bar there.

Heading west from Bratislava, south of the Danube, you soon enter Austria, where bike paths (some along local roads) lead to Vienna and other destinations.  The terrain in this part of Austria is mostly flat.  On the north bank of the Danube just west of Bratislava is Devín, with its picturesque castle ruins, at the mouth of the Morava River.  From here a sealed bicycle path heads north along the Morava all the way to the Czech Republic (about 80 kilometres). Along the way are small towns with pubs and local shops. Unfortunately, there is no cycle path between the city and Devín and the road is narrow in places.  In summer, some buses to Devín carry bikes on weekend services - enquire at the tourist information office for details.

This site (partly in English) has some details about cycle routes around Bratislava, as well as contact details and addresses for local bike shops.

www.opencyclemap.org has most of the routes described above, plus many others, marked on its user-generated online map.