I'm wondering about whether there are any good areas along the Danube as it flows through Hungary that are especially attractive for walking and or cycling, and that might also have some nice historic sites as well. I've heard about some of the different Hungarian river towns and villages along the Danube, but am wondering what might also fit this category.
Google Durnstein to Melk bike ride. It is about 20 miles long I think. But not in Hungary.
You are pushing my knowledge base. I guess we both learn this way.
In town there is Margaret Island http://www.thecrowdedplanet.com/7-things-margaret-island/ but that would only be about 5km if you circled the island. This I have done.
For something longer, organized tours and to rent the bikes you should probably contact http://bikebase.hu/tours or http://www.yellowzebrabikes.com/ or http://hi-bike.hu/guided-tours-routes-by-bike-budapest and see what they can recommend. I also found this from a reseller http://www.budapestbylocals.com/biking-in-budapest.html
The city proper has miles and miles of bike lanes and public bike rentals all over the place. I don't know much about using it but here is some information: https://en.wikipedia.org/wiki/BuBi
Just stumbled on this:
The EuroVelo 6 long-distance cycling route follows the Danube and I have cycled it from Linz, Austria to Budapest. The trail often runs on both sides of the river and you have several options for crossing on bridges, dams and ferries. It is mostly flat, easy riding but some of the Hungarian stretch is not as well marked. Portions between Bratislava and Komarom were gravel with lots of potholes and puddles to dodge. From Komarom to Estergom we stayed on the Slovakian side which was much better. From Estergom to Szentendre was the most scenic and quite popular with cyclists.
Bikeline publishes a detailed, annotated map book of the route in English, which we ordered online. It includes points of interest along the way.
Komarom has one of the most overlooked attractions in Hungary; http://www.fort-monostor.hu/en/informacio/fort-monostor-0 Sorry, the website is terrible, but it a start if you are interested in fortresses.