Are you sure?
It seems a bit basic, but the first thing to think about is why you want to do a PhD in astro. It’s hard by design, and ~90% of people with astro PhDs don’t end up in tenured academic positions. I don’t know if that’s what your aim is, but it’s good to know regardless. So think about whether it will be worth it for you given your goals. There are still plenty of valid reasons to do a PhD in astro when you know you won’t end up as an academic, so just think about it.
What do you want?
The next thing to think about is what kind of research you want to do, and take inspiration from what parts of your MSc you’ve enjoyed or found the most interesting. Theory? Data/observations? Experimental lab astro? Instrumentation? Also consider what topics you find most interesting, and which aspects of them you like. This is probably going to be related to your MSc topic, but if you want to shift fields, now is probably the best time to do it.
Where do you want to be?
Then, one of the bigger things to consider is which continent you want to be on. Each country has their own academic system and way of doing things, though there are some similarities. Keep in mind that some systems are harder to get into if you’re not a citizen or permanent resident. For example, it’s not possible for non-UK citizens to get UK funding to do a PhD there, so you’d need to bring in your own external funds like an NSERC, Marie Curie, Rhodes, or Gates Cambridge Scholarship.
Keep an eye on the AAS job register, “Pre-doctoral/Graduate Positions” section (scroll down). Not all jobs will be listed yet. For example, Amsterdam usually has a few PhD positions available, and they start advertising these at the end of October. For Canadian and American universities, they typically won’t advertise on there, so you’ll want to look through department websites and see what the different professors do. When you find someone whose research sounds like it’d be a good fit, email them to ask if they are looking to take on a new PhD student. Another place to look for potential advisors is in the author list of papers you find interesting! Again, do your homework and read up on them before emailing them. Check out the AstroBetter wiki for lists of programs and advice on applications. But once you have an idea of what you want to do and generally where you want to do it, ask the profs in your current place if they know which places have people doing that kind of research.
For the US, many places still require the GRE and Physics GRE (but not all!), so there’s that hurdle to get over, and you’ll want to email the department chair of grad studies/admissions to ask if they’ll accept your MSc courses.
Tips on the PhD application itself
You will typically need 3 letters of recommendation. Ask your letter writers now if they feel comfortable writing you a strong letter of recommendation. Yes, use the adjective “strong” in there. If you won’t have a letter from your MSc supervisor, you need to have a really good reason, and have another letter writer explain why that’s the case. For my applications I had letters from my MSc supervisor, another professor in the department who I knew pretty well and had taken a class with, and my undergraduate mentor. Get your letter writers lined up now, and when you know where you want to apply, give them the deadlines and the details. Email them a 2-weeks-before reminder, 1-week-before reminder, and a 2-days-before reminder. Also, if there are any discrepancies in your record, explain it to a letter writer and ask them to include it.
Once you’ve written a draft of the research statement that most places ask for, have a PhD/postdoc and your advisor iterate feedback with you on it. It seriously makes it better.
Good luck and have fun!