When sore after a workout, should I rest for a day or push through it?

Rest. Always pay attention to your body. It will tell you when it needs a break and when it is ready to push forward.

It all depends on what you did the day prior, and what you mean by "pushing" through it.

Light aerobic exercise has actually been shown to speed the body's recovery. If you spent the prior day performing intense exercise, it's better to get out and perform some light aerobic exercise instead of doing nothing at all. Keep it light and don't reach the intensities you reached in your prior workout, otherwise, it will take longer to recover.