Purchase a book or chart that will show you the most common bass guitar scales and give you a bit of theory behind how and why they are typically used. Try to purchase a book appropriate for your skill level. If you have been playing for a while but simply never got around to scales--as is the case with many intermediate, self-taught bassists--you can purchase a more advanced book. If you do not want to purchase a book, you can find scale charts online for free, though many players find that actually having a book is more convenient and a bit easier to follow.
Focus on learning the five most commonly played bass scales first. These scales are the major scale, minor scale, pentatonic scale, minor pentatonic scale and the blues scale. Learning these first few scales will not only make you a much better player but they will give you a better understanding of how scales work. This will enable you to learn more complex, more specialized scales in the future with greater ease.
Practice your scales at least half an hour per day. If you need to look at the scale chart at first, make sure it is plainly visible so you don't have to stop and slow down. Practice the scales in the first position until you have mastered them, then move on to the other positions. As with most skills associated with the bass guitar, learning your scales has a lot to do with muscle memory, repetition and note memorization. Without a regular practice schedule, it will take much longer to learn the basics and your knowledge of those basics may be shaky, at best.
Practice your scales by playing music with other musicians. While jamming with friends may be fun, it can also be a great way for people to work on their individual skills in a setting that is both challenging and supportive. Take turns playing scales with a guitarist or play along to a drummer as he practices to a metronome. If you can't play with actual musicians, consider purchasing a CD of backing tracks that you can play along with, or using a drum machine or computer recording program that has built-in songs like Garageband or Fruity Loops.