On the old system;

Type about:support<enter> in the address bar.

Under the page logo on the left side, you will see Application Basics. Under this find Profile Folder. To its right press the button Show Folder. This will open your file browser to the current Firefox profile. Now Close Firefox.

This is your current profile. Copy the folder As Is to a folder on a thumb drive. Then Copy the folder to the new system.

You can copy files from one profile folder on a different computer to your current profile folder to recover/transfer specific personal data.

• bookmarks and history: places.sqlite
• bookmark backups: compressed .jsonlz4 backups in the bookmarkbackups folder
• cookies.sqlite for the cookies
• formhistory.sqlite for saved form data
• logins.json and key3.db (decryption key) for Passwords saved in the Password Manager
• cert8.db and possibly cert_override.txt for (intermediate) certificates stored in the Certificate Manager
• persdict.dat for words you added to the spelling checker dictionary
• permissions.sqlite for Permissions and possibly content-prefs.sqlite for other Site Preferences
• sessionstore.js for open tabs and pinned tabs

See also:

• https://support.mozilla.org/en-US/kb/Recovering+important+data+from+an+old+profile

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

Congrats on the new computer.

As the first two replies describe, your best bet is to copy the data from your old computer to you new one, assuming your old computer is still working or its hard drive is still readable.

If that isn't an option, did you ever set up a Firefox Sync account, or copy your Firefox data to Google Chrome (which may have synchronized with your Google account)?