Subs: lone wolf packs? Just a fleet of subs? Whats a good number?
For subs it depends on where for what purpose you intend to use them. Myself as UK, I focus them all against Italy initially, and then against Japan once I have swept the Italians out of North Africa. As to numbers, you want to divide them up to cover a wide spread of seazones in an entire region. I would suggest 3 as the smallest likely group, and perhaps 9 or so as the largest, depending on how far, how wide, and how deep into contested waters you want to project them. The more (for all three of those) you are aiming for, the more you use. But don't overdo.
Patrol (defending supply lines?): Mainly Destroyers with a couple bigger ships (which?)
Defending supply lines means defending from convoys raiders. That is ASW (anti-submarine warfare) operations. For that, yes, you want destroyers (DD) as your primary workhorses, augmented with a few heavier hulls to protect them in case they run afoul of surface combatants....in the Atlantic anyway. In the Med, where you have to worry about the Italian navy, don't bother with dedicated ASW groups. Let SAGs (surface action groups, aka gun-based task forces) shoulder the burden there, so that you can do double duty of hunting raiders and reducing the Italian navy at the same time. For the IJN (Japanese navy) don't send anything but strong combat forces, including whatever CTFs (carrier task forces) you may have, into contested waters - they just might chew you up otherwise.
Search and Destroy: Destroyers, middle ships, big ones (what ratio? Subs?)
If by this you mean task forces intended to hunt enemy surface combat vessels, then you build these from the top down. Either you build a SAG or you build a CTF, and in each case you begin with the heavies and then supplement them with escorts and screens as appropriate for the level of enemy threat you expect them to face. More on this in the next bit below.
Carriers? How would you use these?
A CTF is optimally based on no less than 2 carriers (CVs) and no more than 5. You then add 2 or 3 strong gun platforms, e.g. battleships (BB) and/or battlecruisers (BC), or 3-4 weaker ones (CA, aka heavy cruisers) to protect them in case an enemy manages to close to knife-fighting range. Finally you add screens, a minimum of 3 times as many screens, DD or CL (light cruisers) as non-screens. So if, for example, you have 4 CVs and 2 BBs as the core of your CTF, they should be accompanied by at least 18 screens.
For SAGs it is simpler. Just start with the core of gun platforms (BB, BC, and/or CA) you intend to use and then again add at a minimum 3 times as many screens.
As to how to use these, it is much the same between them. Send to the ocean region you intend as their hunting grounds, then give them orders to patrol it. If you suspect the enemy may have large task forces in the area, choose a tight formation to concentrate your fleet. If you are hunting smaller groups and don't expect to face any large fleets, you can choose a stance that spreads your forces wider to cover more area but in more dispersed fashion.
Also, how high up the tech tree does one need to go. Do I need to keep up to date or can I skip a few generations?
As UK you really don't need to research any ships at all, if you don't really want to or if your larger war strategy requires the research time be spent elsewhere. You start with enough of a navy to achieve the minimum historical needs of the UK (as does the USN). That said, you do at least want to pursue your naval doctrines, so that those ships you began with can perform well. The fewer new and/or more advanced ships you build, the more you need to be sure you push your naval doctrine.
See above. Naval doctrines are your upgrades, for the most part anyway. If you deploy carriers, then more advanced carrier-based plane variants might be advisable down the road a bit. But don't skimp on your land and land-based air forces in order to get there.
Other notes:
The reasons you send SAGs into the Med to fight the Italian navy and CTFs to fight Japan are fairly simple. In the Med you are in a constricted area, often in range of enemy land based air cover. This can neutralize a CTF's own air power, thus rendering it more or less impotent in the face of enemy surface combatants - very bad. Facing the IJN in the Far East, however, is different, usually well away from enemy land-based air, where your own carrier-based air power can turn things to your advantage.
If you have to focus on just one ship type to research and build at a higher tech, make it destroyers. You will use more of these than any other ship type, and you want them to be both able to find and sink enemy subs and to be able to provide some air defense for your heavier task forces. If you are going to advance two research lines for ship hulls, make carriers your second line and then use the variant mechanism to give them even larger carrying capacity.
