There are a lot of things you can do to break in a new pair of running shoes, like wearing thicker socks or bending and stretching the shoes at their flex points, but nothing can really beat just taking a walk in them. By walking in your new shoes, they can bend and flex with your feet and gradually adjust to your feet's specific shape.
If your shoes fit properly, this shouldn't take too long at all — just a few days. Shoes that don't fit well will take much longer to break in and in the long run, you are probably better off returning them for a pair that fits better.
If you want to break in your shoes while running, it isn't recommended that you go on a long run at first, but try shorter runs until the shoes can be properly broken in. When you don't break in a pair of shoes, the risk of foot pain and developing blisters is much higher.