The first thing to think about is the decrease in friction between the contact patch and the road. This means you'll need to brake earlier, corner at a lower speed and squeeze on the power in an even more progressive way. In general, slow down or you risk sliding off the track. On a one mile track, the difference between wet and dry times can be as much as five to ten seconds.
Approximate coefficients of friction (roads) with a multi-purpose tyre are shown below:
For slicks, the difference is dramatic:
Braking with locked wheels
On dry roads, the stopping distances which can be achieved when braking with locked wheels vs threshold braking are relatively minor. However in the wet, if you lock up the wheels there is a much more dramatic reduction in friction which can lead to you ploughing ahead on entry to a corner. If you do lock up wheels you'll suddenly wish you had ABS, but if not now is a good opportunity to practice your threshold, cadence or avoidance braking techniques.
Tyre pressures and grip
Lower tyre pressures will increase the size of the contact patch, so letting some air out (while adhering to manufacturers guidelines) will allow you to gain a small grip increase in wet conditions. This will only help up to a point though!
Tread depth and braking distance
Recent research at MIRA (UK) measured the stopping distances at 50 mph in conditions that represented moderately heavy rain (0.5mm to 1.5mm water depth). The research discovered that tyres with a 3mm tread had a 25% better performance than those at 1.6mm. This represents an extra 8 metres (25ft) added to the stopping distance in wet conditions. In one test, when a tyre with a tread depth of 8mm was compared to one with only 1.6mm, the stopping distance increased by 13 metres (42.25ft).
So the moral here is if you are considering a separate set of wet tyres for track use, try not to use them for everyday driving and make sure the tread is over 3mm deep.