Can I change the keyboard shortcuts for tab switching in Firefox?

AutoHotkey is the answer. I tried those Firefox extensions, but they can't handle all the actions I wanted. AutoHotkey can. Try something like this in a .ahk file:

#IfWinActive ahk_class MozillaWindowClass
{
   ^Left::Send ^{PgUp}
   ^Right::Send ^{PgDn}
}

The Keyconfig add-on for Firefox allows you to program most tab-selection keyboard binding.

The SwiftTabs addon even lets you assign single key shortcut for tab switching

On OS X, you can switch tabs to the left or right using option+command+left/right. Command+{/} and ctrl+page up/down do the same thing.

I don't know if comparable shortcut keys exist on Windows, though I would guess that ctrl+page up/down work.

Source: http://support.mozilla.com/en-us/kb/keyboard+shortcuts

The Menu Wizard extension worked best for me. Simple and intuitive UI support. You can follow this link for a how to.

Keybinder Addon worked best first for me. More easily than the currently most popular answer.

It trivially mapped ctrl-arrowleft and ctrl-arrowright to switch between tabs left and right (on linux with firefox 45.01).

It seems good to do any keystroke I want.

I will note that, under gnome-linux, the alt key is bound to the desktop (for popping up the command dialog). You can work around that by carefully hitting Shift, then Alt; rather than the Alt first.

Since I had trouble with all the already mentioned plugins I've had to search for another way to do this. I found another Plugin that allows you to customize shortcuts that did worke: The dorando-keyconfig plugin.

1. Install it from https://addons.mozilla.org/en-US/firefox/addon/dorando-keyconfig/ and restart Firefox.
2. Go into about:addons (or open Addons Settings via the menus) and then into the Dorando Keyconfig Preferences.

3. For each of the below, press [Add a new key], add the values and check the Global box:

   Name: tab_next
   Value: gBrowser.mTabContainer.advanceSelectedTab(1,true);

   Name: tab_prev
   Value: gBrowser.mTabContainer.advanceSelectedTab(-1,true);

4. Now you can add a shortcut of your choosing to the keys.

I found those functions in the description of the plugin, which referenced the original Mozilla Forum post which has a link to some Examples.