It's been a long time since I've studied Biology, but yes, single cells are capable of dividing.
Your memory is spot-on. Almost all somatic cells (which is to say, those cells that are not involved in the production of gametes) are capable of undergoing mitotic divisions, wherein the parent cell duplicates all macromolecules (DNA, RNA, proteins, etc.) and organelles, enlarges, and then splits off, portioning up the macromolecules, organelles, and cytoplasm into two, genetically identical (assuming there are no spontaneous mutations or defects in the proteins responsible for replicating the genome) daughter cells.
To make matters more complex, there's always the issue of distinguishing the eukaryotic cell (those that make up the protists, fungi, plants, and animals) from the prokaryotic cell (bacteria and some algae), as well as interactions between those cell types. An interesting method of evolution concerns the "development" of mitochondria (for plants and animals) and choloroplasts (for photosynthetic plants). Ostensibly, a proto-eukaryote ancenstor cell was infected by an intracellular bacterium, creating a symbiotic cell which provided structural and metabolic provisions for the parasitic bacterium, while the bacterium provided molecules such as ATP (ie: energy) for the host cell; after some concurrent replication of the host cell and bacterial parasite, the mitochondrion was, in essence, "born". The evidence that points to such a symbiotic relationship in the modern eukaryotic cell lies in the fact that the mitochondrion greatly resembles a typical, Gram-negative bacterium (not something that I really want to have to explain, as it might just make a boring post more boring; but if you really want to know about the Gram classification and Gram-staining, you can
read about it here). The mitochondrion has two plasma membranes, a single, circular chromosome that comprises its genome, and 70S ribosomes (eukaryotes have larger, 80S ribosomes).
Such an advantageous relationship (gaining the ability to undergo aerobic respiration for a massive ATP payoff, while sacrificing a portion of the cellular volume for "rent") is believed to have led to the modern animal (as well as plant and fungal) cell. I don't see how it would be implausible to develop cell-to-cell cohesive mechanisms (which do exist in the form of structures such as tight junctions, gap junctions, desmosomes, etc.) on the basis of sharing cellular resources; such mechanisms could invariably lead to groups of cells, forming the first tissues. Large aggregations of cells would then need to "adapt" based on their location in the tissue: cells that are in constant contact with the environment might better "survive" if they have increased expression of genes that code for proteins designed to excrete cellular wastes that have accumulated in the other cells that occupy the same tissue, for instance.
*Shrug* I can't really rationalize things better than the established theories can. I'm, after all, just a student. The theories and their dependence on logical progressions make sense to me, and I recognize that they do not answer all of the questions about evolution (as they cannot account for the emergent properties in many organisms - consciousness, self-awareness, and so forth). However, in terms of having systems develop up to the level of a plant or animal on the basis of successful functioning in the environment doesn't seem to be far-fetched to me.
EDIT - If the endosymbiosis theory is of interest to any of you (the theory about the origin of mitochondria and chloroplasts), Wikipedia has a
decent treatment of it.
EDIT 2 - On another note, the cell does carry out digestion and excretion; how else would it have been able to obtain the building-blocks for macromolecules and cellular structures? I like to credit the cell with a lot, but it isn't capable of spontaneously combining elementary particles into the necessary elements. Cells use a series of triggers when they encounter material that lead to it being ingested in a membrane-surrounded vesicle (a phagosome), which comes into contact with a lysosome (an orgenelle containing digestive enzymes and a low, acidic pH). Once the material is degraded, its component molecules and atoms are used in anabolic reactions. Wastes are excreted by a similar mechanism, but in reverse - degradation, packaging into a vesicle, and then budding-off at the plasma membrane to release the waste to the environment. This is also how a number of molecules like hormones and enzymes are released and, indeed, how extracellular digestion in the small intestine takes place; extracellular excretion is quite important.