I trai a kickboxing - karate mix. We got a belt system and it's probably karate.
How long do you need to get your first belt? And how do the belts follow in colours?
You get your white belt on your first day, but I suppose you mean your first rank up. You earn a new rank when your instructor thinks you're ready (usually following a test). Belt systems vary from style to style and from dojo to dojo.
It takes 3-6 months between colored belts and it takes the number of years between degrees of the degree. Example, if you just turned 3rd dan you will wait 4 years to get 4th dan.
You often get your first belt on your first day (a white belt.)
Color progression depends on what type of karate you are learning.
How long it takes depends on the school. If a school is just out to get your money, progress is very fast and you will have to pay for every test. There are also a lot of "in between" belts.
A dedicated school that only award belts when you have really accomplished something will (obviously) take longer.