Each program is different and there are pros and cons to moonlighting. Some programs forbid it, some will only allow within their system which isn't true moonlighting, you're just picking up extra shifts for more money but with staff nearby, and other programs unofficially encourage it such as mine but you must be approved to moonlight which often depends on ITE scores, and how good of a resident you are. Most programs that allow moonlighting, people start sometime near the end of their 2nd year. Also, most start out moonlighting in a safer environment with lots of coverage and low acuity. The pros are that it provides a tremendous amount of confidence in the autonomy, and oftentimes first jobs ask if you've done any moonlighting because it's usually seen as a positive attribute in that you've already been out from under your institutions wing and been able to handle things on your own independently. The negatives are that you don't want to be involved in a lawsuit as a resident. You also don't want moonlighting to interfere with your learning. Someone that graduated from a program I'm familiar with last year and is in his first year of private practice just got named in a lawsuit while he moonlighted as a resident where he missed a DVT in a young pt who subsequently showed up the next day with a PE. You don't want to start your career off this way. The money is attractive, but don't do it until you feel ready. There will be no attending standing nearby to bail you out of trouble. You probably also need to know your ACLS/ATLS stuff down cold and feel comfortable in any code situation.
This is generally information relayed to me by the upper level residents at my program. I have not started moonlighting yet and probably won't until the beginning of my 3rd year.