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It really depends on your content structure, how well you have it organized, and whether you need to rework your content for a new theme.
If you're going with an 'out of the box' solution, and are coming from the same, then there shouldn't be any major issues other than specific theme settings you'll need to configure.
If you're coming from a custom theme then you need to consider that your custom theme was likely tailor made around your content, and your content specifically setup around the theme.
Now, your best bet would be to make a copy of the site on a new subdomain, call it staging.yourdomain.com, for example, which will give you the freedom to tinker to your hearts content without affecting your live traffic. Then, once you're confident in your changes and want to push them live, move yourdomain.com (current live site) to backup.yourdomain.com, then move staging.yourdomain.com to yourdomain.com to put it live. ANy host worth a lick can easily help with this.
As for SERPS and such, changing a theme won't affect any of that so long as you're improving the site, as opposed to hurting it.
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