“We spent the first month just debating what to call ourselves.”

You're a member of a band. You're telling someone the story of how you chose the name for you band. You spent a really long time trying to choose the best name. You describe this period:

We spent the first month just debating what to call ourselves.

spend (time) (doing something)

You use the word "spend" with time because we think of time as being very valuable, just like money. To "spend time ___ing" means to use your time to do it.

debate (an issue)

When you "debate" something, you talk about it for a long time, and try to convince each other of an idea. A "debate" is somewhat similar to an argument, but a "debate" sounds less emotional and more logical.

"Debate" can be followed by a noun:

debate politics

or you can follow it with a clause, like in the example at top.