In chronological order of appearance of the terms:

Genetics ? the study of the rules of heredity ;
Gene - a part of (usually) DNA, that decodes one protein;
Genome ? the total of all genes that are found in one type of living thing. Btw, the sequence of the human genome is completed, now scientists are trying to get useful information out of this code.
Genomics ? a recently born science of studying and comparing genomes. It happens that some genes are similar even in bacteria Eschrichia coli and Homo sapience (us)

As you see all this words are derivatives of Latin Genus and all are international ? they are the same, for example, in Russian only they are pronounced with Russian accent :-)