The Germans started off their campaign by signing a treaty with Russia, this was their cue. Russia was their only problem in Europe, and America was on the other side of the sea. Once the treaty was signed, the invasion opportunities were endless. The Germans wanted to strike at the heart of the Judaism they so hated, so the invasion of Poland wasn't questioned.
The invasion of Poland was in September 1st, 1939. The Attack was fast, the Polish troops were defeated in weeks upon Nazi arrival. The Germans invaded with over 2,000 tanks and 1,000 planes, with this they broke through the Polish defenses almost immediately. Shortly after a lot of heavy shelling, Poland surrendered on September 27th, 1939. This invasion is sometimes known as the Phoney War, or Twilight War, because nobody around the world knew about what was going on in Poland until around the time they surrendered.
After the conquering of Poland, it was time for Hitler to conquer more of the Eastern front, and try to establish a foothold in Europe. He invaded some countries, and forced others to sign treaties allowing him to establish forward bases in their countries. Italy entered the war during this time, and they joined the alliance. Italy was weak during this time; they were in debt and had little to no resources. Their troops were ill trained and had almost no mechanized army, so the transport of the troops was mostly on foot. The army of Italy was big, but compared to the other powers of Europe, they weren't much of a threat.
The greatest mistake of the Germans in WWII was Operation Barbarossa. This operation was named after a Holy Roman Emperor in the medieval times. The goal was to take over at least half of Russia to establish a front there, known today as the "western front". This failed because Russia at this time was (and still is) one of the largest military and economical power houses. The Germans have to spit their forces in two, so this allows for the allied powers to ship troops into Berlin with ease. Hitler commits suicide at the realization of his defeat.