Average growth is 1/2 inch per month, but some people's grows at twice that rate, while others only see a 1/4-inch growth in a four-week span. Your metabolism, diet, and other factors unique to your genetics affect growth speed.
If your natural color is dark brown, and you opt for platinum-blonde highlights, then you'll see dark dots of regrowth in approximately one week. If you're naturally light and closer to a dirty blonde than a brunette, then you might not notice roots for two months.
I've actually had clients ask me many times to leave roots when I'm coloring their hair. I think dark roots can look cool, but gray roots not so much.
Blonde is usually the hardest on your hair's health. That's because in order to lighten hair you have to strip pigment from it, and that requires ammonia and peroxide — two of the most damaging chemicals in hair dye. The more solid your blonde, the faster you'll need to return to the salon. I often recommend blonde highlights, which are more manageable.
Since little-to-no lifting of your natural color is required when going brunette, this color is the least damaging and the most low-maintenance. I almost always use ammonia and peroxide- free dyes when dying hair darker, because the practice involves mostly depositing (not stripping away) color or covering small amounts of gray.
Because red fades the fastest (red is the largest color molecule, so it slips out of hair with every washing), this is the shade that requires the most effort and maintenance. Don't wash your hair until a week after you go red. I suggest only weekly shampoos after that. You can use a dry shampoo throughout the week if you feel oily — rinsing with water is okay, too. And, every other time you wash your hair, use a red-enhancing shampoo to keep your color on point. Be sure to get a recommendation from your colorist to ensure you use the appropriate one.