What to do during the first stages of your PhD program? Fresh from undergrad, I will be starting a math PhD program this upcoming fall in the United States. After several conversations with math professors and current PhD students, I noticed that several people advised having a sense of "urgency" right from the very start of your PhD program. Most recommended working to complete preliminary exams as soon as possible, but there was some variance in their opinions about when to jump into your thesis. One professor recommended "courting" potential thesis advisers by doing research projects with a professor or two during the first couple summers. 

Do you agree with this call for urgency? If so, how do your recommend attacking those first 2-3 years of a (pure) math PhD program? Is it ever too soon to decide on your thesis topic? Recall that I do not have a master's, although I have taken three graduate-level algebra courses.
I'm a third year, and I have found it immensely helpful to get started with research as quickly as possible. 

The thing is, a lot of times you get stuck in mathematical research, especially when you're first starting out. Also, you will have a harder time finding a concrete problem then you'd think. Additionally, you could use your research to guide which classes you take (at least after quals). Lastly, making progress is hard and takes time. This means that the best way to do research is to always have a research problem.

It may not be your main focus, your main focus may be your exams, but if over your first two years you've spent random times thinking about a specific problem (and have geared your studies towards a problem), you can begin to make a lot of headway. So I highly suggest you start the research process. Talk to some advisers. Find something that interests you. Read some papers in your free time. Use this to find an interesting problem. Talk about this with someone who researches the area and now you have an adviser. And just start thinking about it. This is all pretty leisurely and can be done in your "free time" between classes, after classes, etc. Don't worry about the time consuming parts (actually writing down solid proofs), just get your mind churning.

When exams are done, you'll be surprised how ready you are to write what you've come up with. Even if you haven't solved much, you have likely narrowed down a clear path of where you want to go and what you want to prove (and have an intuition as to why you think it's correct). Write down that path as a "homework problem" to yourself, and now you're full-time research.

(Some people will tell you to wait until you can understand your project. Honestly, I didn't understand what I was doing (stochastic dynamics) for about half a year. After finally taking the classes on subjects related to the field, I can say that classes wouldn't've helped much. The only way to get into a research project is to just start doing it, and keep trying different books/resources until you understand all the background. Classes help, but less than you'd think.)