Do not overestimate the amount of time you have with collections.
The best way to make the most out of your interactions with these collections is to keep a detailed research journal each time you handle materials.
Plan out realistically how many times you will be able to visit these collections. You'll find it's not as often as you'd like.
It is very common to brush off a seemingly insignificant piece of material in a box, only later in your research to realize you may have missed something important. You will be left wishing you had taken better notes.
Analyze sources much like we will during our hands on sessions in the collections. Ask questions, decide what you need to learn to contextualize.
When you're able to, take pictures.
Be reflective. Take a moment to synthesize your thoughts after your visit. Write down questions you have or impressions of the collection, its creators.
Ask yourself, why was this collected by UT, or at all? Why do we provide access to this collection? Use finding aids and archivists to learn about the origins of a collection to learn how it fits in with other collections UT has.