Here's a hard earned tip. The first few months, nobody's going to expect much from you anyway. Try to stretch the time you're 'getting up to speed with the code' as long as possible. But in that time, write as many tests as possible. Don't tell anyone unless they ask; if there is no testing culture, they'll think you're just wasting your time on it. Then at some point you will be expected to start delivering value. At that point, you can start making changes (refactoring where necessary) without fear. This will let you grow your status in the organisation until you have a reputation of being the guy that can get things done and doesn't complain about how bad everything is and how everything is someone else's fault when he's assigned a task. At that point, you can start making others write and maintain tests while you take control of keeping oversight of improving the code base quality.

Do not be tempted into 'rewrites' because you're setting yourself up for failure.

It's a long play, yes. If you're feeling the way you describe just from the current state, you're probably not senior enough to do it. It requires much more than just technical skills - those are abundant, it's not what you can build a career on. It's a bit Machiavellan, I guess, but in the end, everyone wins - but you need to have the Vision that others are not able to understand, so you have to find ways to make it happen anyway, in adversarial circumstances.

Good luck.