Repairs and maintenance
Just go to Bicycle Station. I don't care how many bike shops you ride by on the way there. It's an open secret that this is the best bike shop in town. The owner, Mike, actually cares about his customers and personally helps you with whatever you need.
I had a hard time finding a bike shop to sell me thinner tires for my mountain bike and install them, but Mike took care of me right away.
You aren't allowed to ride a bike across MTA bridges. If there's a pedestrian path, you can push your bike across. Of course people ride across anyway, but some of these paths are not designed for it and could be dangerous, like the RFK bridge from Astoria to Randall's Island.
Another consequence of the MTA bridge bike ban is that Google Maps-style apps will not give you biking directions over these bridges, not even if pushing your bike across, is worth it to save time. It's a shame, because Randall's Island has some very nice bike-friendly bridges to Manhattan and the Bronx.
You will see the MTA bridges called Triborough Bridge and Tunnel Authority (TBTA) bridges. They are toll bridges (and tunnels) built by Robert Moses and taken from his control by Governor Rockefeller, a story famously told in The Power Broker.
You can ride your bike over NYC DOT bridges.
The MTA is owned by New York State. The NYC DOT is... owned by NYC.
Training and overtraining
As a sedentary office worker I tend to go overboard on the weekend. It turns out this is a bad idea for cycling and set me back quite a few days where I had to rest and recover. During my recovery I did some research and found that you should track your rides and gradually increase distance, giving yourself a week or so to recover.
I still haven't figured out how to balance weightlifting with long-distance bicycle rides.
Some of the climbs are pretty tough on the legs, especially the climb up Amsterdam Ave around the 130's in Harlem/Manhattanville. With that climb, once you get to the top, you lose all of your elevation in a block or two and don't even get to enjoy the free speed due to stop signs!
The Old Croton Aqueduct Trail is the beginning of the Empire State Trail. I'm looking forward to building up the strength to try this ride someday, especially the parts that are off road. It would be amazing to get on a bike in NYC and just ride to Canada!