Blogger loves changing their theme according to season or just for fun. However, some blogger change their theme and comes all the disasters: Image not loading, excerpt missing and etc.
Here we compiled a few checklist:
Read the documentation

Did you read the documentation before download? Or contact the theme provider to have them check with you with your current theme. Some theme require some setup before fully functioning as their demo. Some, well perhaps you’re not using it, especially post thumbnail.
List down the functions you use

Sound very challenging, but it is actually quite easy to check. How did you update your blog? What input you used? For instance: Title, content area, post thumbnail, post format, tags, category. Check if those theme support those you’re using.
Which version of WordPress the theme support

Seriously, a lot of nice themes online are old, the developers did not update the theme for ages. You’ll have to consider security and compatibility, but make sure you’re using the latest version of WordPress too. Most old theme doesn’t support custom menu, custom header or custom background, so if you use any of those, do consider another theme or engage with a developer for an update.
Backup, backup, backup

Unless you’re using VaultPress. If not, please do a backup before you changing a theme. Anything could happen with just a simple click on “Activate this theme”. You can consider using BackWPup plugin, which is very flexible. You can choose to backup only the files and database, or just database, daily backup, weekly and etc, you get the idea.
It is important to check all those, or more (please point us if you have any idea, would be deeply appreciated) before changing a theme. However, if budget is not a big concern, it is best that you can come out with a more personalised branded theme. Remember, there is no 999 number to call when your missed anything when changing a theme.