I have had the same brand of running singlet for 10 to 12 years. It is light and is made out of breathable material and isn’t made out of cotton. To allow the perspiration to leave your body. For running shorts you can spend a lot of money of these but it isn’t essential. You start spending big money when you want to make a fashion statement. Socks I try and wear two pairs because it ads in cushioning, you don’t have spend a lot on these either.
Decent exercise shoes is where a lot of money can get spent. It is best to know what is best for you. Also if you will be using tech items such as a heart monitor.
In a word layers.
You may want to remove layers once you are warm. I would wear clothes very similar underneath to summer training. Running tights are good to keep your upper legs warm and prevent injuries. During the colder months consider wearing your running tights to bed, this may aid in recovery.
The biggest issue in Winter is finding the balance of being comfortable and warm enough. Or even dry enough if there is a drizzle rain about.
Stretching is also an important part of any exercise routine. Clothing that has a Lycra component that allows you to stretch is the best. If exercising indoors, using comfortable socks on an exercise mat during a stretching routine is fine.
Overall your activewear needs to be suitable and safe to use. After that it is icing on the cake.