Can I run cables into the side of my breaker panel?
The NEC doesn't have a problem with cable entering a panel from different knockouts. In fact that's why they're there. But the entry point must be bushed to prevent the sharp edge of the knockout from cutting and damaging the insulation from the conductor and comes with a strain relief method. This can usually be purchased in any big box or even a local hardware store in the electrical section. They come in various types and manufacturers but the store should have someone there to help you select one.

Hope this helps.