Is my furnace operating normally?
It sounds to me like you've got a two stage furnace.  

Stage 1, low heat, just enough to maintain temperatures.  Won't produce warm air, just warm enough air - much more efficient than running full burn.

Stage 2, high heat, for when it's actually cold.  Produces WARM air intended to raise temps.  Less efficient than stage 1, but gets the house warm.  Stage 1 follows.

The alternative is a 1 stage furnace that's full on, full off, and thus must cycle on and off over and over again.