Been cycling my commute for 2 years now, rain or shine. Planned to be a fair weather cyclist but realised I hated trains more than crap weather 
I already had a bike with mud guards and a basket. It is big enough for handbag, laptop and a change of clothes bag. Personally I prefer to change into dress and tights/shoes when I get to work. That means it doesn’t matter what state you get in en route, just change when arrived. If my hair is wet it goes in a bun or plait on arrival.
I bought nothing at all for the first six months, just wore cheap black Ieggings (not sports) and a top. I had a crap pac a mac cagoule in the bottom of a cupboard that did the job for rain, and then switched to an old snowboarding jacket when it got too cold.
The first thing I bought was a waterproof basket cover and a microfibre towel, but even then it was after a few months not immediately.
I ‘invested’ in some (non cycling) sports legging and a couple of t shirts, plus a hi vis cagoule and lights when I decided I was reasonably serious and going to keep it up.
I could do with some waterproof trainers I guess, but it’s never a priority! It’s not too bad to be honest- which d why I still do it despite living ten minutes walk from the station.