When running sudo apt-get update, I get a number of errors such as this one:

W: Failed to fetch http://us.archive.ubuntu.com/ubuntu/dis ... 6/Packages 404 Not Found [IP: 91.189.91.23 80]

Is there a new repository url that I could use to get the most recent upgrades?

It is "normal" as described by David in this link post :
viewtopic.php?f=4&t=904

dist upgrade is not a good idea
It should be better to update.

I was hoping to update firefox and find a solitaire game program. Is there another way to do this?

For Firefox, do you check those link ? :
https://support.mozilla.org/en-US/kb/update-firefox-latest-version
http://www.computernetworkingnotes.com/ubuntu-12-04-tips-and-tricks/how-to-update-firefox-in-ubuntu.html

Thank you

You're welcome !
For a solitaire game program, check this package of games :
AisleRiot
https://wiki.gnome.org/Apps/Aisleriot
For installing, in Terminal :


For hearts :


If install is OK, you will find games in "Games" directory in the laucher.
Enjoy !