First, I recommend backing up your entire personal settings folder (AKA Firefox profile folder). See Back up and restore information in Firefox profiles. For example, you can copy your entire Firefox profile folder somewhere outside of the Mozilla folder such as your Documents folder. This will preserve the maximum amount of information.

Second, depending on the specific problems you could try one of these:

• Firefox's Reset feature. This duplicates certain key data from your active settings folder into a new one, bypassing some add-ons and custom settings. This resolves configuration conflicts, but doesn't repair Firefox program files. More information in this article: Refresh Firefox - reset add-ons and settings.

• Reinstall Firefox "over" your existing installation. This will repair program files and replace any default files that have gone missing. If you do not choose to remove your personal settings or reset everything to defaults, then as much as possible of your current setup will be preserved.

Problems with bookmarks and history not working properly can be caused by a corrupted places.sqlite database file.

• http://kb.mozillazine.org/Bookmarks_history_and_toolbar_buttons_not_working_-_Firefox
• https://support.mozilla.org/kb/Bookmarks+not+saved#w_fix-the-bookmarks-file

You can try to check and repair the places database with this extension:

• https://addons.mozilla.org/firefox/addon/places-maintenance/