I also want to welcome you to the ASA and the ASA forum. My name is Keith Miller and I am a long-time member of the ASA.
I will try top respond to your inquiry as well as I can in a brief post. I am not an evolutionary biologist or geneticist, but a paleontologist. So I come to such questions from the perspective of the fossil record.
As stated by Randy, evolution and speciation is a population phenomena, not one of individuals. That is because the genome of a species is a population genome that includes all of the genetic diversity within that interbreeding population. The genetics of those populations, and the anatomical characteristics that are expressions of the genetics, change within those populations. The biological species definition is a population definition -- "a population of interbreeding individuals that are reproductively isolated from other similar populations under natural conditions.”
The recognition of modern species is not a straightforward process. There are populations of organisms that form a continuum of conditions from completely genetically isolated, to species populations with limited but non-zero genetic interchange, to subspecies that may interbreed when they come into contact (and form hybrids), to ring species that vary across their geographic range populations but cannot interbreed where the two terminal populations come into contact. The difficulty in identifying living species is significant -- usually species are simply identified by a set of diagnostic anatomical characters and not according to the formal biological species definition.
When defining species from the fossil record, the biological species definition obviously cannot be applied, and all that is left is the suite of observable anatomical features (and an incomplete set at that). Fossil species are usually described as a population of specimens with a certain range of variation around some mean of specific character states. The observed variation within the defined species population should be less than the difference between those traits and those of other defined species populations. If later discoveries show that the defining characters significantly overlap between two defined species, then those species become "synonyms.” Species always imply a certain degree of morphologic variation. Now because of the limits of the fossil record, we may only have a few specimens that have been designated as a distinct species. This happens if they are distinctive enough from all other known previously known specimens. But what may, and often does happen, is that when more new fossils are discovered those species definitions are abandoned or redefined. In a very real sense the vagaries of preservation on the fossil record determine our species definitions because the samples are discontinuous in time and space.
"Punctuated equilibrium” has generally been very misunderstood by the lay public (and even some in the scientific community). It was not intended to imply discontinuity at the level of evolving populations, but rather it was applying the concept of allopatric speciation to the fossil record. Allopatric speciation argues that the most common mechanism for speciation is the isolation of a small population due to some physical, climatic, or other barrier to gene flow. The resulting genetic isolation (or very reduced gene flow) allows the small population to evolve at a higher rate because it has less genetic inertia. Because the more rapidly evolving isolated population is very small, and occupies a small geographic area, it will have little or no chance of actually being preserved in the fossil record. Only once the species population has expanded both in size and geographic extent will it have some change of being recorded in the fossil record. However, population genetics suggests that rapid evolution within large populations in difficult -- unless there is some long-term environmental change that forces adaptation. Thus the fossil record would most commonly record large slowly-evolving species populations, and fossil records of the speciation process itself would be rare. Evolutionary rates are thus variable over time (they are punctuated), and the fossil record is not likely to capture those changes occurring during speciation itself. There are examples in the fossil record, but they are rare.
Finally, it is very important to recognize that named taxa above the species level are abstractions. They are the result of the application of particular rules for classification and grouping of species. Higher taxonomic categories (genus, family, order, class, etc.)are also highly fluid and change as the body of fossil data increases and as taxonomic procedures change. The appearance of a new higher taxon in the fossil record is simply a speciation event -- no more, and no less. There is nothing especially dramatic involved. In fact, two species placed in different higher taxonomic groups may be, and often are, virtually indistinguishable. I discuss this in my chapter - "Common descent, transitional forms, and the fossil record” - in my edited volume "Perspectives on an Evolving Creation.”
All the best,