The movie's plot is predictable and simple. You can also read a regular person's review. But what does a serious cyclist think of it? Well, I'm happy to report most of the movie's pretty accurate. The common bike accidents (even car-bike collisions) are due to the cyclist doing stupid things like riding wrong way, getting door'd, and of course, mixing it up with pedestrians and not behaving like a vehicle. Unfortunately, the lay audience would probably watch this movie and think about how dangerous cycling in a big city is rather than that a cyclist behaving the way the protagonist does in this movie would be an idiot.
There are several points of unrealism in the movie. First of all, the villain of the movie manages to get around the city of New York faster in a car than a strong cyclist can do it. That's impossible in any big city, even one as small as San Francisco, let alone New York. If car drivers could get about faster in New York than cyclists could do so, then bike messengers wouldn't have jobs.
There's also a bike race that's entirely unrealistic. A cyclist with a derailleur bike with brakes could simply draft the fixie and win in a final sprint. The fixie couldn't possibly draft the derailleur bike with brakes because the derailleur bike could brake harder than the fixie could. Of course, both cyclists in the movie are testosterone-filled boneheads rather than intelligent druggies/road racing cyclists, but as the only professional cyclists around who don't take have to performance-enhancing drugs to succeed, one would think that they'd be familiar with the tricks of the trade.
Having said all that, if you could turn off your brain, it's a perfectly enjoyable movie. My parents both laughed and enjoyed the movie despite not being completely versant with English (the movie is very thin on plot, as I said). Watch it for entertainment, but if your mom already worries about you riding your bike, tell her it's a stupid movie not worth her time. You wouldn't even be lying.