Learning braille is a lot like learning print. Neurologists using FMRI scanners have discovered that skilled braille readers use the same parts of their brains (including the vision parts of the brain!) as sighted people do when they read print.
Young blind children are encouraged to explore different textures with their fingers, much as sighted children are taught to recognize different two-dimensional shapes as preparation for learning the shapes of letters. There are storybooks with tactile pictures and braille text. Even if the child is too young to read the book, following along as an adult reads the story can help the child make the connection between spoken language and the writing on the page. A blind child may spend part of the school day in a resource room with a braille teacher, and the rest of the day with sighted peers doing the same reading and writing activities as the other children, except in braille. Some of the blind child's learning materials will be different from those of his/her peers, to introduce the special symbols and contractions used in braille, but most of them will be the same.
Adults who originally learned to read print but lost their sight later in life have much less work to do. They already know how to read, they just need to use a different alphabet plus a few extra symbols. Some people get braille lessons at an adjustment-to-blindness training program, and some elect to use self-study or correspondence courses. The braille skills and reading speed of people who learn to read braille as adults is tied to how often and for how long they practice their skills.