One of his first problems as pope was dealing with conflicts between Gregory IX And Holy Roman Emperor Frederick II. One was that Frederick had been supportive of a Sixth Crusade, but then was lax about taking part. Another was that Frederick had captured some territories in Lombardy belonging to the Papal States. Gregory called a general council to vote to depose Frederick, but Frederick captured two cardinals who were on their way. This intimidated the remaining cardinals, who were reluctant to oppose the emperor afterward. Gregory had denounced Frederick as a heretic (he was such a religious skeptic that Dante placed him in the circle for heretics).
Innocent, in his earlier role as Cardinal Fieschi, was on good terms with Frederick, but as pope he had to continue the policies of his predecessors, demanding the return of lands in Lombardy. Frederick refused, of course, and his continued political attacks on papal rule created enough of a hostile environment in Rome that Innocent became concerned for his freedom. He snuck out of Rome in disguise in 1244, making his way ultimately to Genoa. A few months later he went to France, winding up in Lyon where he was warmly welcomed.
In December of 1244 he summoned his bishops to the First Council of Lyon; the goal was to minimize Frederick's authority. It was the smallest general council ever: many members feared Frederick's wrath and did not attend, and bishops from the Middle East and Far East were hampered in travel by (respectively) Muslim and Mongol hostilities (see here and here). (Innocent's attempts at dealing with Mongols shortly after would fail.) The council excommunicated Frederick, throwing Europe into turmoil until Frederick's death in 1250.
With Frderick's death, Innocent felt safe in returning to Italy. He also doubled down on the idea that he hd the right to interfere with secular politics. He appointed Afonso III in Portugal. He helped Henry III of England buy a title in Italy, even though Henry had been giving trouble to Archbishop Edmund Rich.
In other news, Innocent formally approved the Order of the Poor Clares, named for Francis of Assisi's friend. (In the picture above, he is granting charters to Franciscans and Dominicans.) He reversed earlier popes' orders to round up and burn copies of the Talmud, being convinced by a team of rabbis that the Talmud was a foundation for them to be able to understand the New Testament.
His time as pope has been woven through this blog for years, and it was high time he got his own titled post to bring some of these references together in one place.
Speaking of things that get mentions and might deserve a fuller explanation, the Papal States have been mentioned above, as well as here and here. Let's explain what they were and how they got started.