How would a plant sprout and grow in a zero gravity environment?
There have been several experiments in growing plants in microgravity (strictly speaking, we do not achieve "zero-g" since astronauts remain in orbit about the Earth).

Changes in plant growth due to the influence of a gravity field is sometimes called gravimorphogenesis. More specifically, gravitropism is a differential growth response of plant organs to gravity. For example, roots grow downwards (positive gravitropism) and shoots grow upwards (negative gravitropism) on Earth.

Studies (e.g. 1) suggest that in micro-g, there is no preferred direction; roots may grow "up" and shoots "down". 

It is thought that this growth response is due to the relative distribution of auxin in the plant. On Earth, auxin will preferentially move down into the root-tips due to the location of amyloplasts in the root-cap cells. In micro-g, amyloplasts do not settle at the "bottom" of the plant, therefore there is a more generalized distribution of auxin, and therefore there will be no preferred growth direction.

In addition, changes in plant gene expression as a response to micro-g environments are also being investigated (2) and suggest that auxin transport inhibitors may block the activation of the auxin responsive promoters in Nicotiana spp. (tabacco).

(1) Mechanisms in the Early Phases of plant Gravitropism CRC Crit Rev Plant Sci. 2000 ;19 (6):551-73  11806421  Cit:65 

(2) Transcription Profiling of the Early Gravitropic Response in Arabidopsis Using High-Density Oligonucleotide Probe Microarrays, Plant Physiol. 2002 October; 130(2): 720–728.
doi:  10.1104/pp.009688