Developing a strong relationship with your immediate manager is important but remember:
Managers are not looking for friends; they are looking for employees they can trust.
Your immediate manager has your future with that company in their hands, but you are also a part of their future. You are there not only to do your job and help customers; you are also there to help make the manager look good. The best way to start a strong relationship with a manager is to let them know you are there for them. Make sure they know you are available for projects, that you will help with problems, and that you can be counted upon. One way to do this without seeming pushy is to listen. Don’t eavesdrop, just listen when they talk. What frustrates your manager? What are the problem areas your manager talks about?
And be specific with your responses. Don’t say something like “let me know what I can do to help,” that is too generic. Say something like “I heard you say you needed help with that project. I can do x.”
Of course, your manager is there to help you as well, but be proactive when you go to your manager with a problem. For every problem you have, try to think of one or two solutions to offer. That will tell you manager you care about them.