How do you know if an organism's alleles are dominant or recessive?
The common practice for introductory genetics is to depict dominant alleles in capital letters, and recessive alleles in lowercase letters. Between A and a, you would know the A allele is dominant because it's a capital.
In practice, this is a highly oversimplified approach, and the real answer is that you never really know which allele is going to be dominant or recessive until you put them in an organism and see what happens. Thus the easiest way to find out, from scratch, is to start breeding them.
Genes are not isolated; they "talk" to, and interact with, other genes, and even the simplest traits can be controlled by a half-dozen genes or more. Eye color, for instance, may be controlled by 15 or more genes, a big contrast to the traditional depiction of one gene with blue/brown/hazel alleles. Furthermore, there is no gene "hierarchy" where only the most dominant gene is expressed; in fact, there may be multiple simultaneously expressed genes which all equally contribute to the phenotype. Even the idea of "dominant" and "recessive" is oversimplified, and fails to fully capture the complexity of gene expression.