Any other tips about cycling in the rain would be appreciated
Avoid it if at all possible; it is not fun. Never mind the discomfort of being wet and cold, all your kit will need washed and dried after the ride. Your bike will be a mess; and in the long term, wet weather throws dirt into your drivetrain which will cause premature wear. It will also eventually get into bearings (in axles, pedals, headsets) and rust them. Rim brakes? Watch for extra wear on the wheel rims caused by the grime thrown up in wet weather.
If it still sounds like no problem to the hardy cyclist, make sure you are very visible to the clueless car driver wandering about in their 2 ton tin boxes who haven't bothered to read the instructions for the ventilation system and cannot see out of their misted up windows, further exasperated by the wet conditions and reduced visibility. These people are a liability at the best of times; in bad weather they are even more dangerous to the vulnerable cyclist.
Sorry to put the dampers on it if you were planning to go cycling in the rain, but IMHO it turns an enjoyable experience into a miserable, torturous, ordeal to be endured. Unless you really NEED to cycle, wait for the better weather and enjoy it!