Hi GLADYSMS7, congratulations on the new computer!

Firefox has two different file formats for bookmark transfers:

(1) JSON "backup" format

This is a dense, not very human-readable data structure that may preserve a few more details. You create the file using "Backup" and you restore the file using "Restore". Note: Restore completely replaces whatever is in the destination browser with the contents of the JSON file.

Ref.

• Restore bookmarks from backup or move them to another computer

(2) HTML "export" format

This traditional cross-browser format invented by Netscape in the 1990s can be viewed as a web page of clickable links. You create the file using "Export" and you import it into another Firefox installation or third party browser using "Import".

Ref.

• Export Firefox bookmarks to an HTML file to back up or transfer bookmarks
• Import Bookmarks from an HTML file

I suggest creating both formats "just in case."

Also, you could back up more files if you want to carry over more data. These articles go into the details:

• Back up and restore information in Firefox profiles
• Recover important data from an old profile

You probably will be able to install the current Firefox release (67.0.4) and thus there will be a large version jump from Firefox 52 ESR. In such a case, best is probably to import the HTML backup.

In case you want to transfer more:

• bookmark backups: compressed .jsonlz4 JSON backups in the bookmarkbackups folder
• formhistory.sqlite for saved autocomplete Form Data
• logins.json (passwords) and key3.db (57 and older; key4.db is used in 58 and newer)(decryption key) for Passwords saved in the Password Manager
  you have key3.db, so make sure to remove an existing key4.db and copy logins.json and key3.db
• cert8.db (57 and older) for certificates stored in the Certificate Manager
  you have cert8.db, so make sure to remove an existing cert9.db
• persdict.dat for words you added to the spelling checker dictionary
• permissions.sqlite for Permissions and possibly content-prefs.sqlite for other website specific data (Site Preferences)
• sessionstore.js is used in 52 for open tabs and pinned tabs
  current releases use a compressed sessionstore.jsonlz4, sp make sure to remove an sessionstore.jsonlz4 and also clear the sessionstore-backups folder

You can use the button on the "Help -> Troubleshooting Information" (about:support) page to go to the current Firefox profile folder or use the about:profiles page.

• Help -> Troubleshooting Information -> Profile Directory:
  Windows: Show Folder; Linux: Open Directory; Mac: Show in Finder
• http://kb.mozillazine.org/Profile_folder_-_Firefox