Well, I think first and foremost, the term "do" is a large part of speech and can serve many different roles, especially when you consider the subtle nature (and sometimes not so subtle nature) of emphasis and suggestion, which can vary a lot based on the context of the conversation.
When you use do (or did) before the verb, as in "I did like the song." or "I do like the song," this would usually be preceeded or followed by something else about it. Specifically, "I did like the song." could have the meaning that "I liked the song (but now I don't)." or possibly "(There was something I don't normally like about the song, but regardless of that) I still like it."
The first meaning is based mostly on the past tense, which in this case suggests that you don't like it now. The second example is more reassurance that you like it. However, the first could also mean that given the appropriate context to support it.
These examples:
"I do love~"
"I do worry about~"
"I do understand~"
Seem mostly in a reassuring sense. For instance:
Person A: "Did you like the food I cooked?"
Person B: "Yes, it was very good!"
Person A: "Really? It seemed like you didn't eat much."
Person B: "I had just eaten lunch late, so I didn't have much appetite. I did like it very much though!"
You'll notice in this case, the "did" is used in a reassuring way.
You can use do as a past tense, just be careful because if the context isn't there the person could misunderstand. This is especially true for concepts that could be described in present tense. The easiest example is likes and dislikes. Saying "I did like that band." suggests that you may not like it now, but did in the past.
It can also be used in another way that is technically of reassurance, but is a little different. This could be referencing similar things that you have differing opinions of. For instance:
"I don't like pecans, but I like pecan pie."