Disclaimer: My area is multiplicative ANT, so this answer is largely dedicated to that particular area. An answer that can elaborate on some aspects of additive ANT would be most welcomed.
Question 1. Analytic Number Theory is absolutely an active research area. I would go so far as to say it is thriving at the moment. I am currently finishing up my PhD in the area and can say with certainty that it is a fertile area for research. My area is multiplicative number theory, so I'll primarily focus on this area. Since you did your masters on the Prime Number Theorem, I'll assume you're interested in this particular area as well. To show you that the area is active, I've included at the end of this post a list of 28 researchers who are actively publishing in the area of multiplicative ANT. This list is certainly not exhaustive and mainly consists of the people whose work I've been acquainted with.
Below are a few subfields of multiplicative ANT, along with a paper or two published within the last five years (these are chosen somewhat arbitrarily).
The study of $L$-functions. Since the advent of the Katz-Sarnak philosophy and the ability to formulate precise conjectures (and certainly before this as well), the study of moments of $L$-functions has become a popular topic of research. Other $L$-function topics include zero-density estimates, subconvexity bounds, and pair-correlation results. Some papers include this one by Heap and Soundarajan on moments of the zeta function and this one by Booker, Milonovich, and Ng on subconvexity for modular form $L$-functions.
Multiplicative functions. Quite recently, Radziwiłł and Matomäki
revolutionized (and that term is not an overstatement) our
understanding of multiplicative functions in short intervals (see also their second paper).
Sieve theory. There are many problems which can be tackled via sieve-theoretic techniques. Quite recently, Maynard made substantial progress on some quite difficult problems. See for instance this paper, this one, and this one.
These papers are all research-level mathematics, of course, and I don't include them for you to go and try to learn from at this point. Rather, I hope they illustrate that ANT (at least, the multiplicative variety) is alive and thriving.
Question 2. It depends on what you want to learn and research! That's a pretty broad question, but there are some standard techniques you can expect to learn (I speak from my own experience as a graduate student in multiplicative ANT).
- Summation techniques such as Dirichlet's hyperbola method and Poisson and Voronoi summation are essential and ubiquitous.
- If you haven't already, you'll probably learn the proofs of classical results such as Dirichlet's theorem on primes in arithmetic progressions.
- Depending on your interests, you might expect to learn some some basic sieve theory, such as Selberg's sieve.
- Many problems can be reduced to estimating exponential sums, so you should expect to learn at least some basic forms of van der Corput's method.
It's difficult to say what all you should expect to learn, but these are a few of the things that are standard and/or generally applicable to many problems.
Question 3. As pertains to ANT, here's one thing that comes to mind: ANT (both multiplicative and additive) is a field that can become very technical. You might have an idea about how to make progress on a problem, but the technical details then take 20 pages to write up. A good portion of all mathematics research boils down to hard labor, and ANT is no different. You should expect to do a fair number of unpleasant but necessary calculations. Your coursework will most likely take you through some classical problems in which many calculations are simplified and/or only sketched.
Question 4. Here are some texts that I have found helpful in my studies.
- Introduction to Analytic Number Theory, by Apostol. Complete with numerous exercises at the end of each chapter, this is great introductory text in my opinion.
- An Introduction to the Theory of Numbers, by Hardy and Wright. This is a classical text on general number theory that includes many analytic aspects.
- Multiplicative Number Theory, by Davenport. This is a classic text and a general introduction to the area of multiplicative ANT. It should be accessible to someone with your experience.
- Analytic Number Theory, by Iwaniec and Kowalski. This is nothing short of a bible in the area. It's not particular good for walking through proofs, but it's great to peruse and get a general idea of the various areas of ANT and the methods/techniques used in each. I reference my copy on a weekly if not daily basis.
If you have specific areas you're interested in, I can recommend other texts that are more focused on particular areas.
A most certainly incomplete list of active researchers in multiplicative ANT
Andrew Booker, Vorropan Chandee, Sary Drappeau, Kevin Ford, Peng Gao, Ayla Gafni, Andrew Granville, Adam Harper, Harold Helfgott, Peter Humphries, Rizwanur Kahn, Youness Lamzouri, Xiannan Li, Kaisa Matomäki, James Maynard, Micah Milonovich, Lillian Pierce, Paul Pollack, Maksym Radziwiłł, Igor Shparlinski, Kanann Soundarajan, Terrence Tao, Lola Thompson, Jesse Thorner, Caroline Turnage-Butterbaugh, Aled Walker, Matt Young
It should be noted that many of these researchers are also active doctoral advisors.