Renters insurance is a necessity. Your landlord has insurance only on the building. You are responsible for insuring your belongings. Get renters insurance as soon as you move into your apartment. If you are attending college, however, check to see if your parents' policy will cover you.
Consider a replacement-cost policy. In the stolen bike example, a replacement-cost policy would foot the bill for you to buy a new bike, similar in quality to the one you lost. Without this, you would receive an amount equal to what your bike is worth minus depreciation, which probably wouldn't be enough to buy a new bike. A replacement-cost policy will increase your premium slightly, but it can be well worth it.
There are limits on reimbursement for expensive items. If you have computer or stereo equipment or costly jewelry, you may want to insure those separately. If you have items of great sentimental value, they obviously can't be replaced and you should put them in a safe deposit box.
Your insurance should also cover personal liability. Most renter's insurance policies will cover all non-auto accidents, including accidents that happen away from your apartment. Let's say that before your bike was stolen, you accidentally hit a pedestrian with it and he or she sued you. You are protected against lawsuits because you have renters insurance. Although coverages vary, $300,000 is standard coverage for personal liability.
Keep your tapes, photos or written list somewhere outside of your apartment - at a friend's residence or with a relative. If you have a fire, you don't want these records destroyed as well.