Look Before You Leap
Before you begin to string a bass, look at how the bass strings are already installed on the bass.
Notice how the strings are wrapped around each post.
Pay attention! The most common mistake people make when stringing a bass is wrapping the bass string around the wrong side of the post.
Each string should run in a straight line all the way from the bridge to the tuning post (green arrows good). The string must not angle away from the nut (red arrows bad, very bad). If your strings angle away from the nut, you may break your nut or cause tuning problems! So pay attention to which side of the post each string is wrapped. Keep in mind it is not necessarily the same for all the posts. It depends on the design of your headstock and where each post is. Don’t try to make all the keys turn the same way; make the strings run straight when you put them on.
Also notice how many wraps you can get on each post. The fatter the strings, the fewer wraps you can fit on to the post. You don’t want the string to wrap on top of other wraps. It should only wrap on the post. This keeps the string from shifting around on the post which will cause tuning problems. The picture above shows three and a half wraps around the E-string post. You probably want at least two and a half wraps on each post. More wraps can’t hurt as long as they’re not on top of each other. (I find getting many wraps evenly on the post oddly satisfying. I'm a weirdo!)
Finally, examine how the bass strings are at the bridge end. Look at how each string sits in its saddle. Again it should be a straight line from one end of the string to the other.
Look where the ball end of the string is held in place. Most likely it is somewhere at the back of the bridge, but sometimes the strings go through the body from the back side of the body.
If you have a hole for each string on the backside of your bass body, the strings can run through the body out to the front. This helps add sustain to your bass notes. Experiment with stringing through the body or stringing through the bridge holes. You may find your tone is different.