How to do install on subdomain, then make the switch?

So my current site is on 1.3.5 ver4 database is on the fritz. Titles and descriptions are not showing up and I frequently have the sessions database hang up and cause my site to go down. According to my hosting company, they suggest I upgrade my site.

So my question is (and Im not very techie) how can I build a new site with the latest model on a subdomain and then move it to my main url once complete? Is there a way I can get my 20,000 images copied to my new site or should I re-upload them? And last, once I get the new site up, how can I delete the older version and all its files and databases?

Thank you so much for any help!

No offense but it sounds like you need to hire someone.