Yes, knocking out the Allied national objectives is definitely a key part of optimal Axis play! Because objectives get collected based on the state of the board at the end of each player’s turn, you have to be able to take and hold a key territory to shut down each objective.
To shut down Britain, you can take and hold Egypt or Australia. Taking and holding Gibraltar is usually harder, but can work if the Brits build a factory in Cairo and don’t build much navy. You also need to make sure to leave enough infantry in France (along with maybe an AA gun or a fighter) so that Britain can’t trade with you in France. This can sometimes mean building mostly infantry as Germany on turn 1!
To shut down Russia, you can either take Leningrad early and then move into Archangel, or you can just put so much pressure on Moscow that the Allies are compelled to send in foreign units (usually fighters) to defend the Russian capital. Sometimes you can also pressure Stalingrad, forcing Britain to bring in infantry reinforcements from Persia, but that’s less common.
To shut down America, again, you need to leave plenty of infantry in France, and you also need to make sure Japan takes and holds the Philippines no later than turn 2. Taking Wake Island or the Solomon Islands to shut down America’s Pacific Islands objective is a nice addition, but it’s not always a cost-effective use of your resources…it depends what America is doing. If America is going heavy into the Pacific, you’ll be hard-pressed just to defend your starting territory as Japan. But if America abandons the Pacific, don’t let America collect Pacific income for free!