Blog Azeroth

Here's what I'd suggest to make the upgrade as painless as possible:

First and foremost, make sure you're logged in as the default "admin" profile, some stuff doesn't work properly on a different account with admin rights

- Backup both your data files and your Database (obvious but still)
- Restore the data files to your sandbox blog (you have one, right?)
- Disable all plugins
- Revert to the default theme
- Copy the new code, including the default theme
- Upgrade the database
- Verify that your sandbox works (view site in a separate browser window or tab)
- Update then activate each plugin one by one, skip those which are broken
- Verify that your sandbox still works (refresh your separate window or tab)
- Switch to your normal theme
- Fix what is broken (if at all)
- Final verification that everything is working

Upgrading your blog proper:
- disable all plugins
- copy the new files
- If you changed your theme, copy the changed files
- Follow the upgrade procedure for the database
- Verify that your site still works
- Reactivate all working plugins EXCEPT caching plugins
- Verify again
- Activate caching plugins last

That's the long but IMO safe method.