Mostly this depends on your own needs. You aren't going to need a trailer if you won't be hauling kids or stuff around on your bike. Odds are good, on the other hand, that you will want a way to carry a water bottle. You don't have to be riding extreme distances for the water bottle to just make sense, especially on warm days.
The most important accessory, of course, is the bicycle helmet. It's the law most places that you wear one. A good quality bike helmet can also save your life in an accident. It doesn't take much of an accident for a bike rider to have serious head injuries without a helmet.
If you're going to be riding at night, or even dawn or dusk, lights will be another worthwhile investment. You're hard enough to see on a bike in bright sunlight. You don't want to be harder to see when it's a little or a lot dark out. Go for quality. You aren't investing in a bike part, after all. You're investing in your safety.
Comfort is not a bad thing to think about either, especially if you're going to be riding your bicycle a lot. That's where things like water bottles come in. A water bottle holder seems like such a small thing, but it can mean a lot. And if you enjoy mountain biking, an easy way to carry water can be an issue of safety, although for a serious ride you may need more than just what one small water bottle can carry.
Check out your pedals too. Good pedals are metal with teeth to help your shoes grip to them. I remember hating these as a teen, because I wouldn't pay attention and sometimes get whacked on the leg by one. But as an adult I understand the advantages. A good grip between your pedals and your shoes is a good way to keep your feet from slipping, which can distract you and result in an accident.
You'll probably want a good quality lock too. If you know you're going to ride out and bring your bike back home, storing it in a locked garage, maybe you'll do well enough without one. But if you want to leave your bike alone for even a few minutes anywhere, you need a lock. Bicycles are all too easily stolen.
Whether you're looking for a bike to ride in your leisure time or something for a commute to work, you need more than just the bike to make the most of the experience. The right accessories can make your life simpler, make bike riding more pleasant or even help protect you from injury.
