Cycling is difficult in the city, let's get that out of the way first. If you thought driving in a city was nerve-wracking, then try cycling. Basically the same thing but you have no protection if you get hit. On top of this pot-holes, curbs and trolley/train tracks become deadly as opposed to minor inconveniences like when driving.
I've driven most of my life, but when I moved to Philadelphia I didn't bring my car. I spent a month relying on mass transit (SEPTA can suck sometimes) and walking, which did the job until I got a job. Then I realized that my 40-45 minute walk to work could be cut down to a 12 minute bike ride. So it's better.
That commute was pretty devoid of traffic, which gave me a pretty easy introduction to city riding. When I first rode into Center City it was a whole new world of weaving in and out of traffic and trying not to get hit. There were times I would just get off my bike and walk for a block or two to calm down.
So what can you do to make your cycling experience a little less painful? Definitely start by getting a helmet that you like and also fits comfortably. Because let's face it: nobody likes wearing a helmet, but in the end it really helps your peace of mind. Additionally it could save your life like what happened to my date a month ago. Lights on your bike are also a great addition to make your bike safer.
After that just try to stay calm. In my 3ish months of having a bike I've popped/punctured 5 tubes. It sucked. Two of them happened back to back days, but instead of getting angry I tried to keep a level head, and now I know where a lot of the bike stores in Philly are, which is definitely helpful.
Speaking of, try to avoid construction, broken glass and other random street things. A lot of cities are full of construction and broken bottles/windows (or maybe that's just my commute) so trying to avoid it best you can is really the only thing you can do. To date my punctures have mostly been caused by broken glass bottles/construction materials. Who'd have thought that people actually live and do things in cities?
UPDATE - Went back in to edit this after popping my fifth tube, in the rain, at night. It definitely sucked, but I kept a smile on my face and continued to make my way home. Perfect end to a long weekend home...
Update 2 - Was the victim of a hit and run last night. I walked away with only some scrapes and bruises thankfully, but my bike didn't fare as well (which is why I added this update). Bent front fork and maybe front wheel as well as a broken seat and front brakes. I'll definitely be posting shortly about that experience.