Sir,
The website is made with Joomla and I don't see any problem with Joomla regarding site moving to new host.
Regarding Hosting change, you can set English language in Godaddy, so there won't be any language issue.
So, keeping the website in Godaddy, and making changes in Captcha and menu and show you a easy way to change the Pricing text will cost you a $150. (Including Joomla version update).
Moving site to HostGator, make the website in Wordpress will cost you $200.
Make website from scratch in Hostgator will cost you $250.
I can do any of them in 1 just 1 day.
Thank you.