I use a CX bike with touring tyres, mainly for puncture resistance and offroadlite capability (there's a few places where I can use canal towpath or cinder paths as shortcuts or to avoid traffic).
Do either about 6 or about 15 miles each way (can drive part of the way and leave the car at a park and ride). Usually ride in with clothes for the week in a rucksac on Monday morning, leave rucksac in work for the week and bring clothes home on Friday evening. Shoes, wash kit, towel and lock live at work. If I need to take laptop home or whatever the bag is there for me.
Midweek I manage on jersey pockets, saddle bag, frame pump and bottle. Don't bother with mudguards or waterproof, I carry a gilet (and in spring/ autumn) pair of armwarmers if I'm not wearing them already which is fine for an hour of rain, and I can dry stuff at work if I get wet on the way in.
Bike does make quite a difference IME - the cross bike with slick tyres takes 20-22 mins to do the 6 miles into town, cross bike with knobblies about 24 mins and MTB about 28 mins. Definitely have a variety of routes and mix it up to stop things getting boring.
Bike commuting is ace
EDIT - forgot to say, did buy a second pair of shoes for commuting ages ago but I never wear them. I'd far rather wear damp SIDIs that fit like foot gloves than the heavy, flexy trainery things I bought for commuting so that was a waste of money.