If we're talking seriously cold or icy then a lot of cyclists will train indoors. That's not always possible for some folks for different reasons. I know a few who are gym members and they do spin classes, etc. If you can train at home then the list of turbo trainers you can get now are endless and because of internet connectivity you don't have to train alone. Personally, now I'm retired, I try wherever possible to get out during the day and so far I've not missed too many outdoor rides due to poor weather. A few times I've had to train in the garage but I only have a basic turbo. However, I also have rollers and I tend to use those with a YouTube video to relieve the boredom.
If you have the right winter clothes then riding outdoors isn't as bad as you'd think. You have to be sensible and safety is always high on my list of priorities these days so if the roads are obviously iffy then it's a garage session. It all depends on how keen you are. Some folks will go out whatever the conditions. If riding a bike becomes a habit, like smoking, then you'll ride come what may indoors, down the gym or outdoors.
As for kit....winter jacket, Roubaix bib tights, bib shorts, Helly Hansen top, thick socks, winter boots, overshoes, under helmet balaclava, lobster Sealskinz gloves and if it's really cold a shell jacket to go over the winter jacket just to keep the wind out. Tyres...in the winter I use Vittoria Zaffiros - cheap as chips. Winter bike is old school 7 speed, alloy frame, carbon forks.