How do I protect aircraft carriers especially against submarines?

I usually deploy an aircraft carrier loaded with 8 stealth bombers. When loaded, this aircraft carrier is very expensive, so I need to find an effective way of protecting it.

The documentation says:

TIP: Guard Carriers with a handful of fast-moving ships and a battleship, as losing a fully-equipped Carrier is VERY painful and expensive.

I usually deploy a loaded aircraft carrier with 4 accompanying battleships, one of which is always on the same tile as the carrier to protect it from direct attack. This gives good defense against enemy ship attacks because:

• The accompanying battleships can get rid of enemy units before they even attack, and
• The loaded stealth bombers can get rid of enemy units before they even attack.

Unfortunately, the aircraft carrier is still vulnerable to these enemy attacks:

• Submarine attacks - I cannot see the submarines in advance, so I have no way to get rid of them before the attack. The enemy could deploy 2 submarines in a suicide mission to get rid of 1 battleship + 1 aircraft carrier + 8 stealth bombers.
• Missile attacks - The enemy could successfully strike the battleship + aircraft carrier using one or two missiles.

How would you recommend better defending aircraft carriers from stealth submarine and air attacks? Other than battleships, what other accompanying units should I consider?

Usually, the strategy with fast-moving ships (i.e., destroyers) is to have a few "sweep" the area each turn for submarines, since they can move more tiles than the carrier. This only works perfectly if you have enough to see every tile, but you can at least give yourself much better odds even if you can only see most of the tiles with the destroyer group.

This is one of the areas Civ is pretty realistic - a proper convoy protects you pretty well from submarines, just like they did in real life.

As far as cruise missiles, they're a bit harder to deal with of course - you can put up a 'screen' of destroyers, making it harder for them to get around you, but that's going to cost you a lot of ships. AEGIS cruisers are better for that, as they get 5x defense against missiles.

I usually won't use many battleships if my opponent goes the submarine/cruise missile route - too easy to lose them. Stick with destroyers and cruisers. Get sea dominance before you send your carriers out, and you'll have an easier time of it.