Before you have access to the Spider Ball and Space Jump, you can actually use your Bombs to reach certain areas. In order to do a proper Bomb Jump, drop a Bomb like any other normal time. But the split second before it goes off, drop a second bomb quickly. Leave one Bomb at the top of your jump arc, and then, as you land, the first Bomb will actually propel you to the second Bomb that will further propel you upward.
In order to do this, you need to time your button clicking quickly and carefully. If you're having trouble, time your tapping for the alarm sound your suit makes when you are just about to faint. By learning this pace, you'll swiftly learn how to master the Bomb Jump.