Did animals evolve from plants?
See this paper "Divergence time estimates for the early history of animal phyla and the origin of plants, animals and fungi" for information on the divergence estimates (I'm not sure if there are more recent papers discussing this).

Plants, animals and fungi are eukaryotes, distinct from eubacteria and archaebacteria, which are prokaryotes. The difference being in the composition of the cell, particularly a nucleus contained within a membrane for eukaryotes, along with other membrane bound organelles, e.g. chloroplasts. They all share a common ancestor, according to this paper, that split 1.576 Bya (billion years ago) +/- 88 Mya (although it states the relationships are unresolved - it is often difficult to resolve relationships so deep in a tree). They form distinct groups known as Kingdoms under Linnaean based biological classification; the Fungi, Plantae and Animalia. Thus, in answer to your question, no, animals did not evolve from plants.

Plants have chloroplasts in their cells, which provide the ability to produce energy via photosynthesis. It is thought that the chloroplast resulted from a symbiotic relationship between early plants and a cyanobacteria in that they both relied on each other for survival and so coevolved. Animals don't contain chloroplasts and instead contain an organelle called the mitochondria (although most plants also have mitochondria), which is also thought to have been a bacterial endosymbiont, probably related to rikettsias. 

Protists also contain chloroplasts. The protists are intermediate between all three groups and have been notoriously difficult to classify, being placed into a fourth Kingdom, the Protozoa, although this grouping has been contested. The current Cavalier-Smith system was proposed in 2004 and classifies life into 6 Kingdoms.

Chloroplasts are thought to have evolved from a single endosymbiotic event in Archaeplastida, although there are evidence to suggest some secondary endosymbiotic events. Check out this paper for more information; figure 1 shows the relationships between the different groups and the endosymbiotic events. The Opisthokonts are the origin of the Fungi and Animalia kingdoms.