Goebbels could go quite a few ways because historically he adjusted his beliefs and policies a lot as time went by. In his younger days, he had a bit of an infatuation with Russia. Essentially, he was one of those who believed Stalin was purging Bolshevism of its 'Jewish influences' and that the 'young people' of Russia and Germany should stand against the 'western plutocrats' and fight for 'true socialism'. He stopped advocating for that after Hitler asserted his total control of the Party. He enthusiastically railed against 'Jewish Bolshevism' and 'Asiatic barbarian hordes', though he also supported a compromise peace with the Soviets after things turned ill. Of course, now the Russians hate the Germans, excluding crazies like the Aryan Brotherhood. It might affect his policy in the occupied territories though.
He never dissented from Hitler, but seems to have been uneasy about Hitler's foreign policy in 1938-1939 due to seeing it as risky. Essentially, he wanted the same as Hitler, but was wary about a big war breaking out. So I actually don't think Goebbels would be a total warmonger. In OTL he started advocating total war when the tide turned against Germany - and as a way to gain power since he had been eclipsed in the previous years. Goebbels feelings on the old elites were not pleasant, so I could see purges. Perhaps a cultural revolution.
Goebbels detested Göring, but his relationship with Speer wasn't straightforward. They did clash over the use of resources after Goebbels was made Plenipotentiary for Total War (Goebbels was supposed to provide manpower for the army, Speer wanted manpower for his factories, so both ended up in a turf war), but both were equally radical when it came to mobilising the German economy and people for total war, no matter the cost. When Speer became sick in 1944, Goebbels expressed concern in his diaries since Speer was an important partner. Both also tried to undermine the 'Dreierausschuss' (Committee of Three), which was de facto chaired by Bormann.