Place them in a sunny location, and a cool one if you have it. Perhaps outside in the daytime and inside at night?

I have them in front of an upstairs window facing east. We keep the upstairs at 60 - 65 degrees. Do you think that will work?

I'm a first-timer at seedlings myself. I have read that you can put them in a sunny window after they have 2 leaves (or use grow lights). Pinch them when they reach about 5 inches so they don't get too leggy. When it's almost time to plant them, you need to harden them off. This involves setting the seedlings outside for certain amount of time and gradually increasing their exposure to the environment and the sun.
The "Growing from Seed" forum has a great FAQ page that I have been researching. Even though I'm a first-timer, I'm starting 8 different plants from seed - herbs, veggies and annuals.
Good luck and let me know how yours turn out...I'm growing zinnias too!