I have a few objectives that I like to go for with Italy
Achieve NOs as soon as possible which means clearing out North Africa then the middle east
Stacking the hell out of Gibraltar. The allies can only get to your home territories and your NO territories by getting into the Med and its gonna be a hell of a lot harder for them to fight in North Africa if they cant bring ships in to help. Im fine with them landing in Morocco
Push south into Africa depriving UK of precious IPCs hopefully reaching as far as south africa. With a small mechanized force and some air power you can make quick work of the UK african troops
Branching out from the Middle East I like to both harass India and provide support to Germany in Southern Russia.
Once Gibraltar has a fairly big stack I like to branch out and land troops in Brazil, Panama, Mexico or the West indies. Just by forcing the US player to divert resources to this area gives me more time to stack Gibraltar and prepare for defense. This is coupled with helping Germany in the Battle for the Atlantic
My Italy is usually all over the place so the way I achieve this explosive growth is typically by having German planes back up newly taken italian territories with german planes similar to how italy can openers for Germany on the Eastern front