Play Australia, invade Sicily and Sardinia and get bogged down in perpetual fighting over Calabria. That's what I did. I did manage to carve up huge chunks of North Africa and take Rome for myself.
Can I just suggest that CAS and encryption/decryption bonuses are incredibly powerful? Plus paratroopers with certain army doctrines. I love paras. Paratroopers is basically how I took over the Italian peninsula mid-game with such a small military. The important thing is if you want to save time, with the European theatre try assigning aircraft directly to armies rather than manually selecting zones--all you have to do then is insure that airfields become major targets, or you mass build them.
Also, you do know how army widths work, right? That's probably the most important thing. Depending on army doctrine you generally want to aim at 10, 20 and 40 widths. Early game army experience should be used to maximise the number of soldiers you can simultaneously bring into a fight at once.
I actually like creating 10 width columns of soldiers as fast moving, low number divisions equipped with medical tents, engineers, air and antitank and recon, with a strong artillery and mechanized infantry + medium tanks for 'offshoot' campaigns like taking African territory.
I reserve the 20 width divisions for fighting Japan and fighting in the European theatre.
10 Width columns use less supplies, makes it easy to exploit weaknesses in the enemy line, and can run rings against slower moving forces. Creating a pocket salient and cutting off 200,000 enemy soldiers from supplies is doable and just as effective with a fraction of those numbers and materiel so long as you have the pre-requisite divisions. And assuming you can rebuke certain parts of the encirclement breakout attempt through overwhelming CAS support, letting the rest of your forces dig in, they can inflict ridiculous casualties on the enemy.