Italy's main goal is to secure the Mediterranean so it's protected from Naval invasions, so push in libya until you've secured the levant and morroco. You also need to navel invade and take Gibraltar and the islands around the Med.
Next goal is to take Iraq for the oil
Then it's up to you on whether you want to push into Iran to india and the soviet Union (High risk - High reward play)
or just hunker down, protect the med, have some troops to help defend against a D-day and help the axis take out the soviets.
Personally I wouldn't bother with protecting the horn of Africa, too much effort for too little gain, if the rules allow I would puppet Ethiopia (after grinding for army xp) and use their manpower to make some divisions (I use them for coastal defence) and then never call them into the war so the allies don't murder them.
Don't neglect your navy, you will need one in order to defend the med and perhaps a cheeky sealion later on
General tips to remember; Air is king. If you're yellow air, make more planes. In relation to this remember to get some refineries to get synth rubber.
antitank is much more important in multiplayer than single.
People like to trade back, if someone is buying a resource off of you it's polite to buy a resource back off of them.
remember counter espionage, you don't want land you've conquered to just rise up behind your lines
If you're in a stalemate, tanks + Paratroopers can often give you a nice encirclement and break the lines
