Moving my blog to a new server?

Hi there. Sorry if this isn’t the right place to ask this.

I have a blog that I would like to move from the crap free hosting I use to a paid one that is slightly less crap. But, I am worried that this process will delete the content on there (the posts & comments), because as I understand it all of that exists only on the current server. If I reinstall the blog on a new server, what will happen to all my stuff? Is there a way I can ensure that it’ll be safe during the move?

Any help/advice would be great. thanks!