The only tyres I've had 'problems' (broken wrist) with in the wet were treaded commuter tyres. The only time Michelin has let me down.
Make sure you aren't getting water in your bottom bracket (that's the bike's bottom bracket ;-) and headset. Most road bikes have poor (no) drainage in the bb, so take the seat tube out, turn it upside down, and see what comes out. A bb that's always wet will eventually get stuck in there and you'll have to throw the frame away.
A section of MTB inner tube cut into a fat H can be tied around the seat clamp to keep water out at the back.
Same story with the headset... water can persist for ages in there. It will destroy headset bearings in one season, and steel steerers will rust. You can tie a tube around the bottom bearing race in the same way, but although it keeps grit out, it helps retain water.
Neoprene is rubbish. So are wax and dry lubricants. Get the most expensive synthetic chain lube you can find, and clean the thing regularly with diesel. After a wet ride, the chain will rust. Oil or diesel it quick. Change the chain often (1000 miles) if it's used in nasty conditions.
You can also get water persisting in the wheels. A white Alu oxide deposit inside is the giveaway. There's not much you can do about it.
Hubs seem to be more water-resistant (more grease), but then I'm using MTB hubs. The exception is the chain-side rear one. There is no incentive for water to escape, and if the seals are no good (anything below XT or Ultegra), your axle grease turns to bubbles.
If you ride off the road and can fit mud guards (fenders), you can really prolong the life of the drivetrain by fitting a front mud flap. I kid you not.
Accept that you will wear out rims. Grit collects on the inside surface, and then flows onto the braking surfaces when you go through standing water. This grinds your pads and rims down about 10 times faster than normal. Get lots of brake pads, and check for thin rims by overinflating every now and again.
Store the bike in a dry place. Get a hygrometer.
Hope this helps. I live in Wales, I know what I'm talking about.