My commute is also 4-5 miles and hilly.You definitely need gears, otherwise you're likely to wreck your knees. You also need bright clothing and good bright lights to make sure you're seen. I've been knocked off my bike by someone who said 'sorry mate I didn't see you'. It's very painful.
Warm gloves. A thin sweater to go under your waterproof jacket for the colder days.
If you're riding on paved roads get a hybrid or road bike. A mountain bike is too slow and knobbly tires give less grip on asphalt.
Used is better if you're on a budget, but take advice so you don't end up with a lemon. If you take to commuting the chances are that within a few months you'll have a better idea of your needs/wants and want to get another bike. Good Luck and welcome!