Clear the Cache
The javascript libraries and css have been updated and this can effect both the admin and frontend of your site giving some unpredictable effects. To resolve this you need to clear your web browser cache (not joomla cache) and reload the page. This ensures that things like modal boxes and checkboxes are working correctly.
Database Fixes
In some circumstances the one-click upgrade is not able to apply the small database changes. But don't worry Joomla can resolve this easily. All you have to do is click on the Database tab in the Extension Manager and the changes will be applied.
New Plugins and Modules
Sometimes some of the new plugins and modules will not be available in your control panel. The database fixes should resolve that but I always think it is better to be safe and not to assume anything. Go to the Discover tab in the Extension Manager. Click on Purge Cache on the toolbar and then click Discover. If any new extensions are listed you should tick them all and then click on Install.
Note if their were any plugins discovered then they will probably be installed as "disabled" and you will need to go to the plugin manager to enable them if required.
HTML Text Filtering
The text filtering options have been moved from the article manager to the global filters. As a result you might need to check that any filtering options that you set are still active. In addition, and most importantly, you should go to the text filters and save them, even if you are not changing anything. This prevents an unfortunate instance where all html is stripped from your article when you save it.
Frontend Javascript
On many sites people are using scripts to disable Mootools from loading on the frontend of their site. This may have been done by an option in the template or via a plugin. Due to a change in the way captions.js is loaded on the front end of your site if you are not loading mootools then other javascript may fail. IF you observe weirdness on the front end of your site and you have already cleared your browser cache and joomla cache then re-enabling mootools will resolve this.
What Version of Joomla Am I Running?
In the footer of your admin there used to be a notice with the exact version of Joomla that you are running. Due to a very low level security issue this is no longer there. If you need to check which version of Joomla is installed you can find this information from the System Information menu. Don't forget that there is now an icon on your control panel telling you as soon as your install of Joomla is out of date.