Switching Theme – Easiest Transition

So I know that switching themes can affect how the content is displayed, but I’m trying to keep my headache at a minimum. Just wondering what advice you guys can give me from experience.

Would it be best to copy off all the data from the ftp and create a local site and fix all content display issues on the local version…then upload the changes back to the live site? I can’t afford to have my site down for a long period of time so I’m trying to figure out a good alternative.

Any advice would be appreciated.