It is good to find a group of students to study with, but be wary that some students will "explain" things they know nothing about. So be sure to be discussing specific theorems and lemmas with a textbook handy. Be sure not to be over eager to explain things yourself. It is also important to familiarize yourself with the faculty in the department. We have quite a few older students and visitors around. It is best to seek out a faculty member when you are really stuck on a question. It is alright to ask someone if you can have their name and write it down so you can look them up later.
As you discover an interest in some fields more than others it is important to speak to a faculty member in the area for advise. Our faculty are listed by research area here. You can email to ask advise about which written qualifying exams to take and what books to master for those exams. It doesn't hurt to get advise from a few faculty members and if you find one particularly accessible they may be someone you'd like to check in with more often.
Remember the focus of your attention in the first year of graduate school is preparation for the first exams and faculty can be consulted with help in this direction. Do not start reading articles and doing research until you have proven your background is thoroughly understood with three passed exams.
Save the sample exams to do in an exam setting! When you think you are prepared for the exam, take a sample exam in a timed situation. Then study up what you've missed and repeat. You should not be learning the material by doing the practice exams. Textbooks are organized and their problems are just as hard but in a setting where you can learn from them. Just imagine the calculus student that tries to study for the calc final by taking practice calc finals instead of doing a few homework problems from each section. Remember: if you just took the course and weren't mastering a text at the same time, you probably don't know the material.
At many universities, students must pass their first exams within one year, with September as an opportunity for a retake. These are universities where students finish in five years. Depending on your background, you may need to take more than a year to finish all three exams but keep in mind that these are the first exams and there is a lot more work to be done after they are completed. Students planning to switch into finance should take real analysis.
The focus of all your attention before passing the first exams should be the first exams, not seminars, not reading articles, not special topics courses, not language courses.
Arrange your schedule so that you are at the GC on the same day as your field of interest's seminar. This way the faculty and students you are more likely to work with will be around.
All graduate students who have completed the written qualifiers should be attending one seminar or more a week. Look up keywords in the abstracts before attending. Try to follow for the first 15 minutes or more and then feel free to discretely do homework.
There is more info about choosing advisors at YMN as well as below. Do not delay. It is better to choose too soon and switch advisors than to linger without one for too long. You can also mention to a faculty member that you are looking for an advisor and that you'd like some advise about the field and who to work with. You can visit with this advisor fairly often before making the advisorship official.
If a faculty member seems relucatant to take you as a student, ask if there is some way you could impress them by reading a specific book and demonstrating the knowledge. Some advisors are willing to train students in their subfield with a few independant study courses while others prefer students who have already mastered the basic texts. Some of our faculty with projects ready have posted advertisements for doctoral students at our faculty ads webpage.
Be honest with your advisor. Ask him or her what textbooks you are expected to know and then admit if you have not studied the book thoroughly yet. It is not productive to let your advisor give you articles to read that you have no hope of reading. Meanwhile, an advisor can be helpful when you are still studying texts and you can ask them questions about the problems you are trying and show him or her the problems you think are correct as well.
Ask your advisor worthwhile questions that exhibit independance. Do not go to your advisor or another faculty member to ask for the definition of some term. This should be looked up yourself in a math dictionary or even on wikipedia. You can go and ask about the intuition behind some definition after demonstrating that you know what you are talking about by writing the definition on the board.
Follow your advisor's advise and don't start your own side projects like reading an article that interests you without checking with the advisor first. Finishing a doctorate is a fulltime job that takes years, side projects that are unrelated to your research or future jobs will only delay things.
Make regular appointments to see your advisor and have clear goals as to what must be accomplished between meetings. Do not avoid your advisor because you aren't keeping up with the work. Keep seeing him or her regularly perhaps with smaller monthly goals. The doctorate is your degree and the goals and meetings are for you to pursue.
The Mathematics ArXiV has eprints of research articles which usually appear a year or two before publication. This is an easy free resource to use from home with an excellent search engine. Beware the authors are not seperated out carefully as they are on mathscinet.
For published articles, check the author's webpage or just google the title, before bothering to go to the library. In addition to the CUNY library, there is the New York Public Library on 34th and Madison which has extensive on reserve holdings. Courant and Columbia have their own math libraries.