The short answer is: yes on occasion they did fight along side one another. Initially it was in Northern Africa where Italy sought to broaden its land holdings, and later in the Balkans and the Russian Front.
The German commanders held their Italian counterparts in contempt for the most part, and both sides often disagreed vehemently on tactics. In fact, the Italians felt the Germans were talking down to them when it came to discussing battle plans.
Field troops on the other hand, had great respect for each other. The German troops and attached field commanders saw that the Italians were pretty much just doing what they could with what they had to work with. Most of their equipment was antiquated.
In the air German and Italian's fought together over Spain. During the Battle for Britain Goering very reluctantly allowed the Italians to fly a mission over London on what was basically the last day of the campaign. Il Duce sent 15 twin engine bombers escorted by 70 open cockpit biplanes.
All or Nothing: The Axis And The Holocaust 1941-43
Hitler's Italian Allies: Royal Armed Forces, Fascist Regime, And The War Of 1940-1943