Crossing the line
First out of the gate does not necessarily mean first to cross the line. The line, in this case refers to the finish line. The first horse to cross it, even by a nose, is the winner.
In other contexts, to cross the line can have other meanings. For instance, someone who is guilty of cheating or illegal behaviour may be said to have crossed the line. In this case, the line is between right and wrong, or legal and illegal. Commiting plagiarism, getting involved in vandalism and drunk driving are all examples of crossing the line. When such lines are crossed, it may not be possible to go back.
It is also possible to cross the line socially. To engage in inappropriate innuendo or tasteless behaviour that offends others is to cross the line. This may mean a number of things: public drunkenness, offensive language, and inappropriate and unwanted touching are some common examples.
On the whole, once the line's been crossed, it can take a lot more than a simple apology to return to the status quo.
To cross the line can also mean to cross the border. For instance, we might say we're going to cross the line and do some shopping in Bellingham. With this usage, no negative connotation is involved.