To unlock a new zone, you need to beat 7 levels in the current zone. So, the fastest way to get this particular achievement is to set the difficulty to Casual and play the first 7 levels in each of the first five zones. When you beat 5-1 through 5-7 (any difficulty, any grade), you will unlock the 6th zone, and this achievement will pop.
I think it's worth noting that 5-2 is a difficult level, even with a guide, and you may want to beat 5-1 and 5-3 - 5-8 to get this quicker.
However, if you're going for 200 GS, you should get this achievement as you beat the levels on Expert to get the A+ scores. Specifically, you'll get this just before you get
Check out my A+ expert solution for 1-1 through 5-7 here: http://implode.4shared.com/
The pictures are named for the project and difficulty. P1-01X means "Project 1-1 on eXpert."
If you have any questions or have any trouble duplicating my results, please let me know!