I've started an Italian campaign, and my naval strategy so far is
1. Don't build subs
2. Build naval bombers
But what ships should I build? Some updated destroyers, and possibly one or two aircraft carriers? I certainly won't be able to match the UK's fleet, let alone the Americans.
Carriers can wait at first, and may not even be needed until 1942 or so.. Sicily, Rhodes, and Sardinia are your aircraft carriers. And carriers require a considerable investment in research better used on naval doctrines and surface ships.
Along with NAV build FTRs to control at least the central Med. Air superiority nerfs carrier op efficiency, total air superiority prevents carrier ops. After those types CAS, then Hvy FTR, then TAC in priority depending on available MIC and oil. The heavies are good against the strat bombers sure to be heading your way from England.
Upgrade your fleet. You have a lot of ships but they are old. More destroyers are a must, they die in droves in surface combat. Light Cruisers as well. So upgrade to Level II asap after the 1936 construction techs and Level II when you can. And do not forget to build transports, You will need them.
You start with Level II BBs and Level I CAs and BCs. So which way do you want to go? You have time to get Level II cruisers but you don't start with much NIC, you have to choose one path. Cruisers mean weaker ships but more ships fighting. BB's mean fewer but tougher ships. The objective is to get into a gun battle with the odds in your favor. Due to the current naval mechanics I prefer BBs as they are more likely to survive to fight again.
Radar is vital. It improves both your air and naval detection and interception chances. Level 2 on Sicily, Sardinia, and Rhodes is usually enough at start. if you are short on CIC Sicily is the priority.
Pick one port and build 5 positional AA on it. The Allies love to port strike you, this will help reduce damage. I usually go for Taranto as the northern ports are too close to the UK. Also fly some air superiority over the port if you have the spare fighters, it will stop the bombers from flying port strikes. Unfortunately I have found I never have enough fighters as Italy, so the P-AA is a fall back.
Pre-War Strategy - you may want to annex Yugoslavia before WT hits 25%. That is a lot of CIC and plenty of open slots for more MIC and NIC. Once WT hits 25% the Allies will block any further expansion against neutrals and democracies with guarantees. Some people say also grab Hungary, I disagree as that takes a ton of divisions off the table that the ai needs vs Russia. And Hungary is likely to offer those division to you. Better they die in your service than you get a few more factories as you will get almost no manpower from non-core states.
Tactics - do not sail your fleet until France falls. Period. There is no need to lose ships to the soon to be Vichy Fleet. Use every NAV and CAS you have to attack in the Central Med, preferably with FTR cover. Let them chew on the RN and soften it up. If the Allies try to invade shift your aircraft there. You won't hit transports but you will hit anything giving the Allies naval superiority there.
Only engage when you have air superiority and preferably when you have the odds in your favor, you want a gun battle. The objective is to take small bites out of the RN and live to fight another day. Destroying half of the RN while it wrecks the entire Regia Marina is a Pyrrhic victory you can't afford.
Your subs are best used with your fleets. They are going to die anyway so you might as well hope they sink some of th RN as they die.
Obviously grab the Suez (very easy) and Gibraltar asap to make the Med your lake and allow you outlets for trade. Oil you can grab in the Middle East past Suez, for rubber you need to either build synth factories or trade with Japan. I prefer opening the Suez.