Learn English – The meaning of ‘changing fortunes’

• change schools

This is a common phrase and can be used conversationally.

• transfer to another school

This is slightly more formal, and might be a better term to use when communicating with the registrar's offices of the two institutions.

• go to another school

This phrase will work – there is nothing wrong about it – but it could mean something more temporary and less permanent (such as a one-semester exchange study abroad).

• move to another school

This phrase could work, too, but I probably wouldn't use it unless I was moving to another city. (I probably wouldn't say this if I was going to start going to a different school in the same city as my first.)

In short, in informal conversations, like the one you mention in your question, any of these will work:

By the way, maybe I'm gonna go to another school.
By the way, maybe I'm gonna transfer to Sanders University.
By the way, I'm thinking about changing schools. I'll be attending Sanders in the fall.

And, if Sanders University is not located where you live now:

By the way, I think I'll be moving to Sanders next year.