Talk with your university's admissions office for the name of the person you high school transcripts should be sent to. In addition, get the address of the admissions office, the person's office phone number and whether there is a deadline for mailing your transcripts. Write all of this information down so that you can give it to your high school's records office.
Call your high school's records office or stop in and request that your high school transcripts be sent to your university. Provide your high school with the name and contact information for your university and notify them of the deadline, if any, for sending your transcripts. Unless you are still enrolled in high school, you may be asked to provide some form of identification before your records can be released. You also may be required to pay a processing fee by your high school, so have a form of payment ready.
Follow up with your university's admissions office and confirm that your high school transcripts have been received. If not, contact your high school's records office and ask if they have sent your transcripts yet. If they have not, remind them that the transcripts must be sent to your university and give them the proper contact information again.