Many runners get red-faced no matter how much their fitness improves. Everyone handles the heat differently, and just as some runners sweat more than others, some get red faces and others don't. You may find that you get more beet-red if you're doing an intense run or running in hot or humid weather.
During the warm weather, try to run early in the morning or later in the evening (or indoors if it's really bad). Also make sure you're staying hydrated during your runs and drinking when you're thirsty. Try pouring water on your head, neck, and under your arms to cool off.
If you're experiencing other symptoms besides a red face, like dizziness or nausea, you may be suffering from a heat-related illness. Stop running immediately, sip some water and get in the shade. Although a red face while running is usually harmless, you should still mention it to your doctor, especially if you're brand-new to running.