You should first of all understand that you will need letters of recommendation from faculty. Make a point of talking to faculty, attending office hours with questions. If an instructor is to support your application, then the instructor needs to be able to say something other than you earned an A or B in the course. It is not a good idea to ask an instructor for a letter of recommendation if you did not earn an A or B in the course.
Give the instructor at least two weeks to write a letter of recommendation.
Provide the letter-writer a copy of your resume. You will find a “sample resume” posted at http://math.arizona.edu/ugprogram/mcenter/careerassistance.html for you to use to create your own resume. Career Services also has lots of resume resources.
Minimize the amount of work that a letter writer has to do. If a paper copy is required, then provide the addressed envelopes to the letter writer.
If the letters are to go out in email, then send the email addresses electronically to your letter writer in such a way that he/she can easily cut and paste the email addresses into a message.
Follow-up with the letter writer. A quick thank-you and update on whether you were offered the job, accepted into the graduate program, etc. is always appreciated. Remember, you may find yourself asking this person for additional letters in future, so it's not a bad idea to stay in touch.