The most common advice is to change shoes after running in them for 300 to 500 miles. Most shoes need to be replaced before they look worn out in order to reduce the risk of running injuries. When you buy new shoes, take your old ones with you and show them to the person who fits you. The old shoes, especially the soles, say a lot about the way you run and can help you find a well-fitting new pair. If you spend a lot of time exercising in athletic shoes, it is worth your time and effort to go to a specialty store for runners. A good salesclerk will help you find the best fit, and not just steer you toward the most expensive shoes.
Running shoes do need to be broken in. You should run easily (not speed walk) and short distances only as you get comfortable in your new shoes. Then go for it!