The side lengths of a triangle are increased by 50% to make a similar triangle. Does the area increase by 50% as well?

Yes it does.

For example if your side is 4 and your height is 3 your area would be 6

With the increased length your side would be 6 and height still 3 and your area would be 9

The are increased from 6 to 9 so it increased by 50%.