I would say... apply for a job and see what happens!  We've certainly hired people where this is the most recent thing they have done, they've come straight from completing a BA, sometimes an MA as some universities offer this sort of qualification at MA as opposed to BA.  So this is a good point to enter the industry.  If you do any non-translation jobs first they'll certainly have transferable skills that you can use.

You can also consider applying for an internship as well, if there are any on offer. There are sometimes internships on offer at RWS and there are other companies that do this as well.

I myself, I applied for a job after completing my Masters degree, in translation, and I was working as a translator.  Before applying for a job as a project manager I remember I did an online course as well to get an idea about project management and the sort of things I could expect, and that helped me.  

Bonnie Sandall said:

You can also consider applying for an internship as well, if there are any on offer. There are sometimes internships on offer at RWS and there are other companies that do this as well.

That would certainly give you more hands on experience before applying.