Instead of using a grow lamp, will placing my plants under a regular fluorescent or incandescent lamp work?

It might, depending on the light. But it probably won’t be as effective. Different stages of plant growth require different light spectra. The initial growing stage requires a blue spectrum of light, whereas the later "flowering" stage is usually done with red–orange spectra. Manufacturers of grow light bulbs design them to product a specific light spectrum range and lumen output to create an ideal growing condition for starting plants.