When it comes to literary analysis, the sky is the limit. In other words, there is just so much out there. Let me give you some ways to become a more sophisticated critic.
First, if you are dealing with texts in either ancient or foreign languages, it is important to learn these languages. For example, if you are dealing with Plato, then it is necessary to be able to read classical Greek.
Second, it also also good to know something of the historical context in which the text was written. This will give you an idea of what was happening during this time and offer to you a historical perspective.
Third, there are so many theories out there that you can employ to look at texts from different perspective. For instance, you can use the theories of power from Foucault to look at the dynamics of power in a text. Or you can use the various social theories, such as how ideology works, and trace other nuances of power. Or you can use the insight of structuralism and try to find binaries. As you can see when it comes to theory, there is so much that you can do. A good rule of thumb is to read enough theory to possess new tools to read texts from different angles.