How do I access the first value of a multiple value text field using replacement patters inside a rule? For example, lets say I have a list of objects each stored as separate values of a text field. How would I access the individual values? I have tried using delta and I have tried using entity token but nothing has worked. I hope my syntax was just a little off.
Have a look at the tutorial about "Using Entity Reference in Rules". Starting from about 2:50, and up to about 4:15, it shows how in a Rules Action you can access the 1st, or 2nd, or ... value of a multiple value field, provided you also included a Rules Condition like "Entity has field" (related to the multi value field). So if you have a content type with a multi value field with (say) myfield, then you'd use node:myfield:0 for the very first value of that field (or node:myfield:1 for the 2nd value, etc).

However, the same tutorial, starting from about 4:20 and up to about 6:40, also explains how to loop through multiple-value references, using a Rules loop, and which I think is more appropriate to answer your question.  If you're not familiar with such loops, then I strongly recommend to also have a look at the tutorial titled "Lists and loops".