Best practices for changing weblinks?

I'm managing a webshop where they just added categories left and right and I know want to move them around so the customer can find products more easily.

The problem is that the CMS automatically changes the links to the categories. So if we have:

foo.org/cat/
foo.org/toys/

And we want to move toys to cat we end up with:

foo.org/cat/toys/

Then we have google adwords with a dead link and google queries that needs to be updated. And maybe a lot of dead links from other sites.

What are the best practices? Should I contact the CMS provider and ask them for 301 redirects to be implemented somehow, probably costly?

Are there more aspects that I need to think about? Share me your tips please.