Exporting data for a new server

I am planning on exporting my data for a new server, but how could I do that and how would I import the data?

Would backups do the trick?

Yes. If you check the box to save uploads, that should have everything you need to copy Discourse to a new server that uses the standard install. (You might need to carry over other configuration if, for instance, you have a proxy server or somesuch.)