Is it possible to take classes at another university?

I’m currently attending a college in my home state of Virginia, but I would like to take classes at a different university out of state for the summer or fall. I don’t really want to just take courses at a community college either. How would I go about doing this? Do I have to enroll in the other school or can I just sign up for the classes? Can the credits be transferred? Money is not an issue here so I’m not concerned with living arrangements, transportation, etc.

As a first step, Search on your school’s website for “credits taken at another university” or something like that.
Also find out if the class would transfer ahead of time. A college might have BIO 101 but your college might not accept it. I would talk to my advisor about getting pre-authorization.
I do not think you would have to enroll in a school for summer classes.

For example:

https://www.jmu.edu/biology/faq/transfer.shtml#summer
Can I take courses somewhere else over the summer and transfer the credits back to JMU?

Yes. Summer transfer credits involve the same procedures describe above for transferring credits. They can only be transferred from an accredited academic institution.

Can I take any of the core courses (BIO 114, 124, 214, 224) at another college/university in the summer and transfer the credits back to JMU?

It is not possible for matriculated students to transfer credits for BIO 114 and 124 once they are attending James Madison University. This applies to both majors and non-majors. It is possible to transfer BIO 214 Cell Biology and 224 Genetics and Development. The courses must be approved individually through the Biology Academic Coordinator (Bioscience 1028F). Courses that do not have a pre-requisite of College Chemistry will not transfer for these courses. The courses must have laboratory experiences comparable to BIO 214 or 224 to be approved.

http://registrar.vt.edu/registration-multi-brief/FAQ_Transfer_Credit.html
Q: I am a currently enrolled VT student and I want to take a course at home over the summer. What do I do?
A: Students are required to fill out a pre-authorization form prior to taking courses at another college to transfer to Virginia Tech for credit (this includes education abroad experiences). Contact your academic advisor for information on the pre-authorization process and to fill out the appropriate form.

Some schools offer exchanges in the U…S. Sort of like a semester abroad, but in the U.S. see if your school does this.