By people I assume you mean the average citizen? Then, no, they were not aware this was the start of WW2 until the next year or so.

The German–Polish Non-Aggression Pact of 1934 signed on January 26th was sought as a method to improve relations between the two countries. Prior to this point, there was a lot of questions about the newly formed Republic of Poland (1918-1939) and its size. At the time, it was the 6th largest country in Europe.

Part of this agreement was to weaken ties between Poland and France. Germany and The Soviet Union thought that Poland may be an aggressive nation, and as a result on May 5, 1934 Poland renewed the Polish-Soviet Non-Aggression Pact.

A trade agreement was signed between Germany and Poland later and is believed it granted Germany to settle the eastern border. After Czechoslovakia was invaded in 1939, Germany requested that Poland allow Germany to annex the city of Danzig (Gdańsk), a semi-autonomous city-state, so they could build an extraterritorial motorway and railway between East Prussia and Germany while accepted Poland's post-WWI borders. Poland obviously said no, and Germany invaded Poland stating they were reclaiming their land in early September, 1939.

I know in history class when I went to school we didn't hear more about the invasion of Poland, but quite a lot of happened. Germany didn't just walk into the capital and say, "Hey, thanks for the land." The Polish defenses fought back quite hard. Citizens were kept in the loop about the war happening through newspapers, much like how we rely on blogs or website for news now. The Soviet Union joined in the attack in mid-September stating they were protecting Ukrainian and Belarusia. Polish troops retreated to Warsaw and lasted until late September.

My grandfather was one of many Polish citizens that were sent to work in prison camps during the war. This was a question I asked him once and he said they were unaware of any Germany aggression until the invasion. However, he does recall seeing newspapers about the German and Soviet invasions, and the eventual declaration of war on Germany from other countries. This was before they really started to push arresting citizens and throwing them in prison camps.

Time magazine coined the term World War II. See here on September 1, 1939 just a few months after they issued a magazine that dubbed the previous war as "World War I". In an odd way, they predicted the next war.

To add to /u/waspocracy's analysis:

Two days after the start of the German invasion of Poland, France and Britain declared war on Germany. This date acts as the traditional start of the Second World War. Naming a clear start to the Second World War is problematic, however, as A.J.P Taylor noted as early as 1961 in his The Origins of the Second World War. The war between Japan and China had already begun, and Fascist Italy had invaded Ethiopia a few years prior. Had the historiography of the war been dominated by either China or Ethiopia, surely the "start" of the Second World War would not have been September 3, 1939.

Even then, it is not clear that this should have been marked the start of the war. After all, the British and French showed only nominal military support for Poland after declaring war (a French expedition penetrated only a couple of kilometers into Germany before retreating) and the Germans did not attack France until 1940. This period between the "start" of the Second World War and a direct confrontation between the military mights of the Allies and Germany--a period marked by several naval engagements and the German invasion of Norway, but little else--is known as the Phoney War. World War Two had begun--but it would have been hard to tell for the average Frenchman, Brit, or German.

The British series World at War contains a telling anecdote from a British journalist. While traveling the French Maginot Line (a system of defensive structures built to withstand a German direct attack on France), the journalist asked a French solider why he was not shooting at German soldiers they could see standing across the field. The French soldier explained that the German soldier seemed alright and wasn't doing anything to warrant being shot at. This was after the official start of the war. Not even French soldiers yet fully understood the implications of this "war," however. The Phoney War ended suddenly in May of 1940 after the German army invaded Belgium, the Netherlands, Luxembourg, and France.

As Martin Gilbert addressed in his The Second World War: A Complete History (2004), Hitler also misunderstood the true implications of the British declaration of war on Germany. Hitler hoped that the British were really posing to save face in light of their commitments to Poland and France. He in fact hoped to negotiate a peace with Britain after knocking France out of the war. Only after he realized that Britain would not yield did Germany begin plans for Operation Sea Lion (an aborted plan to invade Britain by sea) and the bombing campaign in Britain.

So to answer your question: no, many people were not aware at the time of the invasion of Poland that the Second World War had begun. This became much clearer after the Battle of France, and even clearer after the start of the Battle of Britain. Nevertheless, war had officially--by declaration--begun just two days after the start of that invasion.