How do I know if I was a full time or at least half time student? I graduated undergraduate in December 2023 and began graduate school in August 2024

Your institution will decide what is considered a full-time course load for the degree program you are enrolled in. If you complete at least half that number of hours, then you will be considered enrolled at least half-time. For practical purposes, the institution will indicate your enrollment status for the half-time test on the Form 1098-T in box 8.

Why do you want to know.  There are several tax attributes that are affected by full time student status.  Eligibility for the Lifetime Learning Credit (LLC) is not one of them.

The biggest issue is whether you can be claimed as a dependent. 

There are two types of dependents, "Qualifying Children"(QC) and Other ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, student status, a relationship test and residence test.

A child of a taxpayer can still be a “Qualifying Child” (QC) dependent, regardless of his/her income, if:

1. He is under age 19, or under 24 if a full time student for at least 5 months of the year, or is totally & permanently disabled
2. He did not provide more than 1/2 his own support. Scholarships are excluded from the support calculation
3. He lived with the parent (including temporary absences such as away at school) for more than half the year.

Enrolling in August (Aug-Dec) usually meets the 5 month rule.