Is a point awarded to my opponent if I serve two lets in a row?
Or can I serve as many let serve as I want until one of my services is legal or illegal?
A let in table tennis is any rally that does not end in a score. The "net service" that you described, where the ball on the service touches the net as it goes over, is one type of let. (All of the circumstances that result in a let are listed in the ITTF Handbook, section 2.09.)
By definition, a let does not result in a point scored. Additionally, the service does not change until a certain number of points are scored. As a result, you can serve multiple "net serves" in a row, and each will be a let.