I've gone through a couple!
My first was a 70's/80 Peugeot UO-8 that I got for $24 at a yard sale (didn't have 25 and made a deal), put velo orange fenders on it, cheapie walmart lights and rode to classes and did errands during college on it, ~2-3mi. wonderful bike for the purpose, I'd recommend if you're a road-based commuter, heavy but still fast, great loaded handling, and classic good looks.
That bike fell off with a roof rack and got eaten by a semi, now the Leather saddle (totally worth the weight and cost for the comfort IMHO) lives on an 80's Panasonic MTB I got for $50.
Currently I ride a 7 miles one-way commute in the Lehigh Valley PA, one mile on road, the rest on a mix of rail-trails and singletrack, so i really need a jack-of-all trades and the old MTB fits the bill. My Advice: Get a lugged steel no-suspension MTB that's slightly too big for you. I ride a 17-18" MTB normally, and this 20" basically has a mix of road and MTB geometry for me. Upright enough to commute, be comfy and see easily, but longer and more stable than a regular mountain ride, especially when loaded. They may need some maintenance and love, new cables, chain, a few replacements for bent parts, but they are strong enough to take some abuse, use great modern-ish components and sealed bearings, and dirt cheap on CL. They're the used '90s cars of the bike world, not yet classics, and not new enough to cost much.
For Lights I currently use a Cygolite metro 550 and hotshot rear, I have zero complaints, battery lasts long enough for a full day of riding, and I charge at work anyway. Plenty bright, though I'm not on busy streets or hyper-fast speeds either. My unnecessary desire is for a dynamo hub light, but I don't ride randonneurs or anything that needs one.
Tires Serfas Drifter Survivor Series, 2" tires. Inverted tread is the bomb for me. Best compromise tire out there. Smooth center strip for fast pavement and hardpack rolling, while the inverted tread provides adequate grip on loose, wet, snowy stuff. The only thing that stops me is 6" or deeper snow, when the rear won't have enough traction to push the front through. They are heavy, but I'm not racing, and the thickness means they're that much harder for anything to puncture (no flats yet, 1 year on them, through glass-littered college-town streets, and old rail beds with plenty of loose metal that ate many tires before these.)
Fenders Velo Orange SS Fenders. Hard to find fenders that fit 2" tires well, and these did. Super durable, and work great paired with some homemade Leather mudflaps at keeping my feet dry. Again, I went for heavy and durable over lightweight, these occasionally eat sticks and things and are none the worse for wear.
Racks More Velo Orange, and another splurge like the fenders, but after breaking 3 aluminum rear racks (I guess car batteries and some tools exceed the weight limits a bit) I felt justified. They integrate nicely with the fenders, giving structure to one another, and the threaded holes on the racks let me do things like attach baskets and tools easily, which helps as I am a woodworking teacher by trade.
Panniers Ratty old Avenir Panniers I got ages ago, have some holes in the bottom even but all I use them for is clothes and food, occasionally I'll bag my laptop and add that in, but otherwise its all my hand tools that travel back and forth with me. Someday I'll build some nice light wooden boxes for them and my food/clothes... someday. Someday ALL the projects will be done hah