Firefox keyboard shortcut to switch search engines in search bar

1. Click at the icon
2. Select "Manage Search Engines…"
3. Select the search engine you want to set a shortcut for
4. Click on "Edit keyword…"
5. Give a shortcut, e.g. "g" for Google

Now you can type into the address bar the chosen search shortcut followed by what you like to search. Example: g weather berne.

To switch to the address bar with the keyboard, use the shortcut Ctrl+L. You could also open a new tab (Ctrl+T), because it sets the focus to the address bar automatically.

By the way: this works for nearly all search fields on any website. Just right-click on a search field and select "Add a keyword for this search…" (don't know the actual English translation). Instead of adding this search to the search widget, Firefox saves it as a bookmark.

In newer versions of Firefox, the shortcut you are looking for is Alt + Up/Down.

There is a shortcut for exactly this task: from the search bar, press control + Down(Next)/Up(Previous).

You can use Ctrl+K or Ctrl+E to focus on the search bar and then use Ctrl+↑/↓ as per Akronix's answer.

Unor's method in conjunction with Ctrl+L is of course a lot faster for everyday use.