Evolution is a big topic and covers all levels of biological organization: molecules and genes, individuals and populations, and every taxonomic level from subspecies to major kingdoms. For this reason the discipline is often subdivided into macroevolution and microevolution, both of which deal with evolutionary processes, but at different ecological and temporal scales.
Microevolution includes the processes that occur at and below the species or population level, like changes in allele frequencies, genetic mutations, natural selection, and genetic drift. Macroevolutionary processes include everything that happens at the level of organization above the species or population, and at longer (geologic) time scale. Macroevolution is concerned with things like adaptive radiation, the tempo of evolution, and the origins of major groups.
Is speciation a micro- or macroevolutionary process? Both. It is impossible to study speciation at only one level of organization, as it involves processes that occur at all levels of biological organization. Genetic mutations, changes in allele frequencies, genetic and gametic incompatibilities, and changes in population traits all contribute to the speciation process. Speciation is responsible for major evolutionary patterns over huge geologic timescales—the evolution of major phyla, adaptive radiations, and the generation of the world's insane biodiversity.
It's amazing to think that all the world's diverse organisms share a common ancestor, but it's true. If we could retrace our evolutionary history back to the very beginning, before there were mammals, plants, or multicellular organisms, there was LUCA, the last universal common ancestor. There may be 20 million species currently alive today, but we all arose from one LUCA.
In trying to classify and organize the world's biodiversity in an understandable way, we turn to evolution to provide us with a framework. Modern taxonomy classifies organisms based on their shared ancestry. The graphical result is a phylogentic tree that shows the relationships among organisms and the nested-ness of groups. Species are nested in genera, nested into families, grouped into orders, and so on. The diversity of the world can be traced back to the ancestors that gave rise to it.
Reconstructing a phylogenetic tree is really reconstructing the history of speciation. We often know very little about the evolutionary changes that occur within lineages, and often even less about extinctions, so speciation events become the major feature of phylogenetic trees. While speciation events lead to novel species and species traits, ancestral features are often preserved and passed on such that biologists can infer information about evolutionary relationships from shared features. Through a long history of ancestors and descendants, we are all related on the family Tree of Life.