First and foremost, it should be stressed that Budapest is one of the friendliest cities in the world in terms of bicycle use (has ended up at the No. 13 spot, according to Copenhagenize Consulting Company). The cycling paths network has a total length of 200 km, while the Hungarian drivers show great respect to the cyclists. However you should always be careful, since many parts of the network are not in the side of the road but in the middle of it!
Also, you don't need a special effort to rent a bike here. There are many offices across the city where you can get one for no more than 10-12 euros per day usually (plus the deposit for it which may require some extra money, but many owners are keencan be convinced to lower it). Apart from that offices, there is also an extensive public bike rental system. In many places there are stations with the green BuBi bikes. The cost is a bit higher (e.g. about 10 euros for 3 hours) but one big advantage of that service is that you can take or leave these bikes almost everywhere. Currently there are over 70 such spots in Budapest. Besides we are talking about a city that in general is very cheap, so there are no other significant costs.
A route may well start from the big Lipótváros park of the same named neighbourhood, in the middle of which stands a monument to Soviet War. Heading southeast, just a few minutes away, the is one of the most iconic buildings in the city, the Cathedral of St. Stephen, that took its name from the first king of Hungary. You will easily recognize it since it's almost 100 metres high, making it visible from well far.
Following Károly körút street, one has the opportunity to see more of Budapest's major attractions. Just before the junction with Rákóczi avenue you will find the huge building of the Jewish Synagogue, which is the largest in Europe. On the same street you can also visit the National Museum if you are a history lover. And of course don't forget to pass by the Heroes' Square, one of the most emblematic in Central Europe.
Of course, having a bicycle offers you the opportunity to visit two of the most beautiful spots in Budapest in my opinion, and spent some time there. I am talking about the two largest open air areas, the City Park (or Városliget) and Margaret Island. I think they deserve a special mention, since you can explore them with a bike from the beginning till the end. The first one is more than just a single park, as it includes from a zoo to a castle, and is a perfect destination to rest before continuing your excursion (tip: try the fresh lemonade in the café next to the lake).
In order to reach Margaret Island the best route is beside the Danube river, passing by the characteristic building of the Parliament. And if you were impressed during the day, you can't imagine how amazing is after the sunset. Margitsziget as the locals call it, is perhaps literally like a miniature of the Hungarian countryside. Small children, families, young people, tourists and many others find a few moments of relaxation in a huge area full of playgrounds, sports facilities and dozens of colors. It is so huge that walking probably will take you many hours (and comfortable shoes), and you can understand why I like it so much, so for me the bike is the best solution. And with it, via the last one of the so-called seven Danube bridges you will get to Buda.
Apart from the homonymous Buda Castle that offers the most amazing view of the city and hosts from churches, chapels and museums, the other half of Budapest is also full of spas. And if you do not already know it, this city is considered to be one of the best (or the best) spa destinations in Europe. Therefore I would highly recommend to hop off your bike and enjoy a rejuvenating time there.
And after a long and tiring day, which was certainly worth it, it's probably high time to return your bike. But before doing that, it would be a good idea to visit the covered market of Budapest, where one can find from clothes and souvenirs to traditional Hungarian dishes, like the famous red chilli peppers or gulyás.
I was able to do most of these activities by bike, when I visited Budapest in June 2014. I believe that it is a fantastic place that you should definitely visit if you get the chance. Most of its sights are easily accessible by public transportation or on feet, but I think it's one of the few big cities in Europe that is so beautiful and you can see it on a bike. For this then reason, if you ever find yourself there seize the opportunity!