hi weepingwillow... welcome to the forum and the wonderful world of commuting

13 miles one way is quite a long way, not impossible by any means. If you find there and back too much effort in the beginning you could ride one way alternately on different days e.g. Day 1: bike to work, bus home... Day 2: bus to work, bike home... and so on...

Don't worry about having all the "right" gear - for every Bike Forum geek discussing "the best" piece of equipment ad nauseum, there are hundreds (thousands?) of commuters getting up in the morning, drinking their coffee and bicycling happily to work on an old clunker who wouldn't have a clue (much less any desire to know) about all the nerdy stuff discussed on this board

You'll learn as you go along - this forum can be a great resource of course - but don't let the nit-picking discussions here put you off...
There'll be set backs along the way, but that's all part of it - makes it all worth it...

Lights is a must... Learn to repair a puncture (carry a spare tube and fix the hole at work/home later)

Clothing? Nah! fwiw I've been riding bikes daily for 40 years or so and have never bought a piece of bike specific clothing... T-shirts, jogging pants, college shirts, running shoes, regular rain gear, it's all good

Have fun!