When my furnace comes on, the fire lights up, then when the blower kicks on, the fire goes out. When the blower is done, the fire comes back on. The blower only stays on for about 30secs at a time, and runs approx. 10 times a hour.

Make, model and year of furnace. Also if you could take a picture of the inside of furnace, and post it here, it would help. Sounds like you may have a honeywell fan/limit control, or other fan control that is causing your problem.