Not sure if you'd like an advice or merely sharing your experience, but after reading your comment I spent an hour or so trying to provide a simple rule of thumb. To my surprise, I failed miserably. But I scraped some info together in the process, so I'll post it in a hope that it might give you a better perspective. Mind you, I'm not a linguist or a teacher.

First of all, I made a list of Russian я/мне lines with English translations next to them just to see if there is consistent presence of a hint in English lines that can point to the right Russian translation. There is none. So if you are trying to figure out the proper pronoun this way, you are doing it wrong.

To make an educated choice between я/мне you'll have to familiarise yourself with nominative and dative cases.

Here is some basic info on the cases in "question" and even more basic examples:

Nominative case of "I" is "я". Nominative case answers questions such as "who?" or "what?".

Кто пришёл? (Who came?) Я пришёл. (I came.)

Dative case of "I" is "мне". Dative case answers questions such as "to whom?" or "to what?".

Дать кому? (Give to whom?) Мне. (To me.)

As you might've noticed, native English speakers won't normally construct questions in a way they are constructed in most Russian cases, so I'd suggest to get familiar with cases and their respective questions first, and try to construct sentences later.

Or, depending on your goal, you can simply memorize common sentences/lines altogether and figure out why they work later.