- Fenders — keep you dry if it's rained recently. I prefer the "full-coverage" kind with a mudflap, but anything that keeps you from getting a stripe up your back is probably sufficient.
- Regular platform pedals (or even better: BMX style pedals) - clips or clipless and frequent stops don't go well together and might mean needing to lug shoes with you.
- A basket or rack — better to put your stuff on the bike than on your back, and you're bound to need to carry something sometimes.
- Some kind of minimal rain gear. There's some cheap packable rain pants and rain jackets out there. You might be able to make do with a rain jacket you already have. Very preferable if the rain gear tries to be "breathable". Check the weather for the day before you leave in the morning and leave at home. Or maybe a packable rain cape...
- Tires — smooth tires that try to be puncture resistant.
- Water — bottle cage (or two) and water bottle (or two). Might want one of those water bottles to be insulated so that you can have ice water on hot days. (I love my travel mug for a hot beverage in cold weather, and that works for cold beverages on hot days, too). Don't necessarily need the water on the bike for every trip, but you should have a way to easily carry water on the bike.
- Pant leg retention — Something to keep your pant leg from getting into the chain. Some bikes have a cover for the chain... You can get a velcro strap... Or you can just roll your pant leg up or stuff your pant leg into your sock...
- Sunglasses — Especially nice if you have some with swappable lenses, because sometimes you want light or clear lenses just to keep the wind and bugs out of your eyes.
- Lights — forward white and rear red. Blinking optional. These could go on your helmet, be removable from the bike or permanently mounted on the bike... Leave them home when you're not going to be riding at night.
- Reflective stuff &mdash If you're concerned about night time visibility, look for retroreflective things. Reflective vests similar to what construction people use are common, cheap, and making your torso visible can be more helpful than making the lower bike parts visible.. Some reflective tape on various parts of the bike. The normal set of reflectors bikes usually come with. Reflective details on clothing, etc.
Any bike you like the feel of riding. Properly fit. A hybrid (flat handlebars, wide-ish road tires) bike marketed as a commuter would probably be ideal, and they often come with everything. However, any halfway decent bike would be fine for what you described, really.
If there's any risk of carpal tunnel or similar issues from your job and/or other hobbies, seriously consider a handlebar setup with multiple hand positions. (bar ends on the flat handlebars, bullhorn bars, drop bars, etc...)
Unless you're particularly into bike repair, forget the on-bike pump and patch kit, carry enough cash for a bus or taxi instead. And keep track of where the bike shops are (and establish a relationship with one or two). Most bike shops can fix a flat in a few minutes without needing to be scheduled ahead of time, and they'll probably only charge $5-$10 (plus the cost of the new tube, of course).
On the other hand: a pump can be mounted to the bike and doesn't weigh much, and a patch kit's pretty small. You'd need a tire lever or two (or multitool that includes them) also. I do carry a pump, spare tube, patch kit and multitool with tire levers, but didn't when I was first bicycle commuting. If you're using a bicycle with for short urban trips and are able and willing to use a secondary emergency transportation option (bus, taxi), flats should be very rare. Changing a flat only takes a few minutes, so unless you get a flat right as the bus is coming it's probably still faster to change a flat than take a bus. And it's reasonably easy...