Just returned from SAP TechEd Amsterdam where biking has always been "the thing" but in traveling to the Middle East and Spain these last few years I see that they too have added bike lanes and "City Bikes" as has NYC. A city bike is a great way to really see a city and also meet with locals. You can get to places that are less accessible to really traverse by foot or car and it saves time ....in Tel Aviv I went along the beach and park drives which were wonderful. Did the same in Barcelona plus went up the mountain to Park Guell. In Amsterdam saw the museums by pedalling through to them plus went through some cool markets. In New York it's fun to cross some of the bridges to other bouroughs that allow bikes (George Washington Bridge, Brooklyn Bridge). In Germany went from the SAP offices in Walldorf to Heidelburg and did that through some forests and villages.
I'm going to do another bike exploration in New Zealand this winter..... lots of places have rail trails where you can bike (tried that in Bermuda too and that was super fun). So here's my biggest tip for those who like to bike. Bring a soft bike seat pad. It really helps cushion the ride and doesn't take up much space. And...it also ensures that you think about renting a bike wherever you are :-).(I forgot mine in Amsterdam but rode anyway). Also good idea (even for men) to bring some hair-ties. They are great for preventing trousers or jeans from getting caught in the wheels. They are also helpful as an addition to the showercap idea.
I love the way the community has added tips. You've mentioned so many I would have shared. I also carry a very small first aid with antiseptic pads, small bandages, minor meds like aspirin and over the counter stomach stuff. I've almost never used it myself, but I can't tell you all how many times I've shared with others who needed.
Also, I love having a moleskin (a small journal) not just for diary like purposes but for travel notes and info that I come upon along the way. I know it can be PDA but I like the paper kind.
And I don't know how many of you need this but I've gotten a passport with extra pages ....And I also make sure that someone has a scanned copy of my documents at home.