As the title states, we just started using the furnace due to the cold weather. Initially if I wanted the heat to get to 70, I would have to turn the thermostat to the max level. Now even this doesn't work all the time.

As a remedy, I just put in a brand new thermostat and changed the air filter. The filter was extremely dirty and clogged. The problem still persists. The heat will turn on for a few minutes, sometimes a little longer but will always turn off again before it gets to the desired temperature.

Please help.

Does the fan come on?

I think so. I just watched it with the top portion off. I heard a blower, then there was a glow, then I saw a little flame, then it glowed a few more times before it shut down. This all took about 2 min.

OK. The system just clicked into gear and was running for a few minutes but there was no flame only the little fire previously mentioned and cold air was coming out.

A glow, then a small flame?
Look at the gas valve. If it is a Honeywell, look for the model number. Does the model number begin with the letters SV?
Regardless of the make or model, please supply us with the model # of the furnace & gas valve.