Safari no longer updates. This seems short sighted of Apple. Should I make Firefox or Chrome my new default browser?

Firefox is best, in my opinion. Chrome has too many poorly tested extensions, one of which I found led to MacKeeper. I immediately removed Chrome.

Firefox would be the better choice: it's reliable, frequently updated, and still works with 10.6.8. It doesn't use Apple's Webkit (which is a bit crashy even on Mavericks), and sandboxes websites so that if one crashes it won't bring the whole browser down. Chrome may have issues with OSX. Apple are not interested in keeping Snow Leopard or its components up to date - the system no longer gets security updates, which means there is an increased risk. They expect you to upgrade to the latest system, though many people don't want to do that (Yosememite has its own collection of issues.)

Link to Firefox