What should I do testing for an old application?

I got this Ruby on Rails application with incomplete testing, (many
controllers or views do not have unit test at all).

Should I go back to write all unit test for each of controller or
view, then do integration test? Or I just forget about all of them,
keep going ahead with my new functions with my new unit test?

My concerns here are: I also need to upgrade Ruby and Rails, or re-
factoring the program. How can I test to make sure the whole
application will be running just fine after all the upgrading and re-
factoring?

Thanks,
Li

Thank you very much! I was thinking about it, but afraid to waste too
much of my time. Your response confirm my thinking. Great. I just go
ahead do it.