How do I input paying for both undergraduate and graduate tuition in one year? After graduating with a Bachelors degree in May, I started my graduate degree in August.

Just enter the 1098-T information. Since undergrad and grad are in the same tax year, both qualify for all education credits so enter them like always. 

Next year, you will select graduate school and only qualify for the Lifetime Learning Credit.

Both the graduate and undergrad expenses will count for the more generous  American Opportunity Credit (AOTC), if you and/or you parents have not already claimed it a total of four times, between you.  Since graduation year is usually fifth calendar year of an undergrad career, it's very likely you will not qualify.  In that case, both the graduate and undergrad expenses will count for the Lifetime Learning Credit (LLC).