How did we become homo sapiens? It's complicated...

Kalahari bushmen, blue-eyed Scandinavians, the Yamomani of Brazil, the Inuit, Japanese businessmen - despite the huge differences in appearance and culture that have developed over the millennia, they are all members of homo sapiens sapiens, anatomically modern humans, the only sub-species of homo sapiens still in existence.

In fact, over the past 20 or 30 years, the idea of a homo sapiens sapiens sub-species has been all but abandoned, and we are now all just homo sapiens. In the same way, it used to be the custom to show Neanderthals as homo sapiens neanderthalensis, i.e. as a subspecies of homo sapiens, but it is now more usual to see homo neanderthalensis as a separate species.

The Neanderthals are our nearest cousins, now extinct. It is not quite clear when homo sapiens originally split from homo neanderthalensis, with estimates ranging from 350,000 years ago to as long as 800,000 years ago, but it was sometime during the Middle Pleistocene epoch, or what we usually think of as the Stone Age. As a nice round number, we can probably say half a million years ago, although of course it did not happen overnight, but was a long gradual process. The Neanderthals, which appear to have arisen in Europe and Western Asia (unlike homo sapiens which probably arose in Africa), died out about 40,000 years ago, probably due to a combination of climate change, disease, and competition with the encroaching homo sapiens population, but for half a million years or so the two species lived side by side.

Another set of cousins, the Denisovans (homo denisova), genetically quite similar to the Neanderthals, also lived alongside Neanderthals and modern humans for a simular period of time, finally dying out as recently as 15,000-30,000 years ago in some parts of Asia. That species, or sub-species, was only identified in the last ten years or so from very few fossil remains, and their exact status remains contentious, but they seen to have interbred with both Neanderthals and modern humans.

So, if modern humans and Neanderthals diverged, say, 500,000 years ago, who were our most recent common ancestors? The simple answer to that apparently straightforward question used to be homo heidelbergensis, possibly via homo rhodesiensis (although many consider those two species to be one and the same). Homo heidelbergensis arose between about 700,000 and 300,000 years ago, but some recent research suggests that homo heidelbergensis was actually just a kind of "pre-Neanderthal", and not a separate species at all.
Evolutionary biology and anthropology appear to be extremely contentious and fast-changing fields, and almost every established "fact" has its detractors and opponents. The delineation between different species and/or subspecies is notoriously fluid. Extensive interbreeding between species, and an inexplicable fossil gap between about 400,000 and 260,000 years ago,  further confuses the picture.
So, anyway, where did homo heidelbergensis come from? Again, there is some confusion and some dispute, but possibly from homo antessor (which some argue is just an early form of homo heidelbergensis, or a European variety of homo erectus), or possibly homo ergaster (now thought by many to be merely an African variety of homo erectus). Or possibly neither. It all gets very confusing. 

However, the generally-agreed progenitor to homo heidelbergensis, and therefore the next back in our direct line of ancestors, is homo erectus, which existed from about 2 million years ago until about 110,000 years ago (as you can see, there are large overlaps between the different species, another source of confusion!) Homo erectus was the first human species to live in a hunter-gatherer society and to control fire, and it may be the earliest member of the homo, or human, genus of the hominid family.

But of course, that would be too simple as well, and there is a case that homo habilis was the first human. Homo habilis was the first human ancestor to use stone tools, hence the name, and provides the link between the (technically non-human) Australopithecus afarensis and the recognizably human homo erectus. Except that there is a move afoot to reclassify homo habilis as Australopithecus habilis, making it not human but still hominid!

Like I say, it's complicated.