How do I unlock new courses in Everybody's golf?

To unlock new courses, you need to both rank up and beat enough VS characters, as seems to explain this user:

The courses (and almost everything else (special shots, carts, swimming, fishing, etc)) unlock as you progress through the ranks (not golfer level) and beat the Vs character challenges. At the moment I'm Rank 6 and I have 4 of the 5 courses available (as well as all 3 tee boxes, normal, long and championship).

So the problem was probably coming from the fact you didn't beat enough VS characters to reach the second course. As for the minimal rank required to be able to unlock each course:

• Rank 3 unlocks the Alpine Mountains course.
• Rank 4 unlocks the Kanaloa Beaxh course.
• Rank 5 unlocks the Vortex Valley course.
• Rank 7 unlocks the Imperial Gardens course.