Does this mean that my first match will be against somebody on my level? So high-level players will never play against low-level players?
It's actually the opposite of that, to start with the highest player will play the lowest ranked player, the 2nd highest the 2nd lowest, and so on. The bracket is structured in such a way that this trend will continue after every round (but only assuming the higher ranked player wins every time). Here's what a bracket with 8 players looks like:
In this example, even though #7 upset #2 and was the lowest seed, the #1 seed still plays the winner of 4/5.
So imagine a bracket like this but on a much larger scale, with 256 players the #1 seed will face the #256 seed in the first round, and so on. In the next round #1 will face the winner of 128/129 (if #1 wins, otherwise #256 will face the winner of 128/129).
For brackets to work you need to have a Power of 2 (2,4,8 and so on); when you don't have that you have byes. X highest ranked players will have byes, the other players will have a match to decide who ends up in the bracket of 256. Once there's 256 players the bracket will continue as normal.
How many players get a bye?
Right now there's 367 players. This needs to narrow down to 256 players. So we need to "cut" 111 players. So the lowest 222 players will play each other, and so the remaining 145 players will get a bye. For that first round, #146 will play #367, #147 will play #366, and so on. As more players sign up less players will get byes, until we get to 512 and then no one gets a bye again.