I just completed the process for the first time, so it is fresh on my mind. Provided here is an easy summary of the exact links and steps that I needed. I went from MediaWiki version 1.20 to 1.22.2.
MediaWiki recommends only transferring files/database between same versions. So, these steps will also quickly include updating the "old" wiki first before installing a new instance on the new domain.
This information is for small, non-customized wikis. For more important data, you should probably read Manual:Upgrading.
Upgrade to newest stable release (optional)
0. Check requirements by reading the release notes
1. Backup files and database. The different ways include SSH, FTP, and via phpMyAdmin. More info
2. Upgrade files. Latest version. One idea is to upload these new files in a different directory, then change the names of the two directories so that the new one starts serving the files.
2.1. Added back LocalSettings.php, Images folder, and custom extensions.
3. Upgrade database. For the web method, navigate to "/mw-config/index.php". Some requirement checks will be performed during the setup wizard process.
4. Test if successful.
Create MediaWiki instance in the new server/domain
0. Backup files and database if you haven't already done so in the first section.
1. Create new instance of MediaWiki on new domain.
2. Import old settings and files from old domain, and change link locations as appropriate.
More detailed info: Manual:Moving a wiki
~ Danial Goodwin ~