It is way more easy than you would think. I’ve done this with Firefox ESR 31.8.0.

1. Run an installer of your Firefox branch on the new computer. In my case, I had an installer for Firefox ESR 24.8.1 laying around. I’ve installed this version on the new computer.
2. Let FF do all the updates automatically. Soon you’ll have the current version on the new PC.
3. Locate the FF profile folder on the old computer. For this, open menu-help-information for debugging-profile folder-show folder. Then close FF on the old computer.
4. The content of your profile folder will be shown in an explorer. The name of the folder will probably something cryptic like ixcc3muk.default.
5. Locate the FF profile folder on the new computer.
6. Close FF and empty the profile folder on the new computer.
7. Copy the content of the profile folder on the old PC into the profile folder on the new computer.
8. Done. This worked like a charm for me. All Firefox settings and bookmarks and all the addons with all their settings are used on the new computer now.

Time needed: Half an hour for finding the solution on the internet, 5 minutes for executing it. Next time I’ll be 30 minutes faster 🙂

Update, Nov 2017:
This way, I could even transfer all the FF settings from a windows PC to a Linux box. Used version was Firefox ESR 52.5.