What to do during summer? So I just finished my 2nd year as an undergrad majoring in mathematics. I plan to continue on to a Ph.D afterwards, but only recently did I realize how behind I was... I haven't done any research, completed any projects or summer programs. I haven't done any outside reading so I don't really know much about any particular subject. Everyone I know in my program has at least something, so I'm worried that I'm really late in preparing for grad school.

I know I should probably consult a professor or advisor, but school's ended already... and admittedly, I've been a little apprehensive when talking with professors since, well, I don't really know anything in the first place, and I know they're very busy, so I eventually just awkwardly ask some questions and leave ASAP having not understood much.

In short, I'm just very confused and don't know what to do from here on out... so any advice would be great.
I haven't done any research, completed any projects or summer
  programs.


That's okay.  You've only just finished your second year of undergraduate studies.


  I haven't done any outside reading so I don't really know much about
  any particular subject.


That can be remedied!  How about taking some summer classes in a community college, in a branch of science that intrigues you?


  I've been a little apprehensive when talking with professors.


You're going to need a bit more chutzpah in order to get your money's worth out of college and graduate studies!  Perhaps you could transition into feeling more comfortable with professors' office hour visits by focusing primarily on visiting the office hours held by grad student teaching assistants for the time being.

It is very common for second year students to be hit hard with an appreciation of how little they know.  Try to ride this storm of self-doubt while reminding yourself of that fact.

It would be great if you could find one or more mentors, either on your own or through a matching-up program.  The mentor might be a faculty member who serves as your undergraduate advisor -- the person who checks your proposed schedule and advises you about what courses to take next year, etc.



Edit 5/20

There are several ways to break the ice.

Ask for some undergraduate advising -- s/he will check your unofficial transcript against your proposed schedule for next semester.  Make sure you have this conversation in this person's office, not over email, because what we want to happen is for the professor's natural mentoring instincts to kick in, so that one thing leads to another.

In the experimental sciences, one can ask for a tour of his/her lab.  You might want to ask your question over at Mathematics Education as well.

Spend some time reading the research descriptions of the faculty in your department, to see whose area intrigues you, then take a look at some of their papers.  When you find one that gets you intellectually excited, make an appointment like this:  Dear Professor So-and-so, I've been reading your "title of paper", and wonder if I could make an appointment to come in and talk with you about this area of research?"  Or something like that.  Researchers love to talk with people about their work.

To get matched up through a formal matching up program, you need to look closely at your department's website, or ask your department administration if there is such a thing.  



During the school year, attend lots of seminars.  Be bold about chatting with grad students over cookies afterwards.  If no one is bringing cookies, juice and paper cups, bring some!