How can I get better at reading handwriting? Are there fonts that look like it, or web sites with examples for practice? Are there some common shortcuts that people take when writing by hand?
I have been able to purchase in China books that have different styles of handwriting including shortcuts. They also have a layer of thin paper over the top of each page so you can trace.
You can also look for books that show common characters written in different styles from print, traditional to script etc.
This is not something you will be taught unless you pay someone to go through with you, as it's not commonly taught in classes.
I don't think that you are taught to write "cursive" in Chinese, most of it is something people just adapt naturally. You just need a really good grasp of characters to understand it. Sometimes people write fast and tend to simplify parts. There is no standard mold for cursive like in english.
But what you can do is read a lot of handwritten documents, just doing that will allow you to see the patterns of how people write.