I don't even feel comfortable connecting through FTP. Am I paranoid?
A little bit, yeah. Websites aren't like your personal computer. You won't "infect" yourself by working on them through FTP.
As far as it getting hacked again, if the site was previously hosted on an insecure server, but it has moved, then the problem has likely been solved as far as that goes. Doing a cleanup job and then creating a new site without the old problems is not really a big issue. Just make sure you follow best practices and use strong passwords and you'll likely be fine.
I used GoDaddy hosting once upon a time, quite a while back. It's fine, if a bit slow. Use strong passwords on everything and you'll be okay.