freeradicals said:
Freeradicals and I - correct.
Me and Freeradicals - incorrect.
Not necessarily. Well, not at all, but first even within the prescriptivist framework this isn't the whole truth. Traditionally the rule has been to use 'I' when you are the subject of the sentence and 'me' when you are the object. So 'freeradicals and me' would be quite correct in the object position, eg:
they laughed at freeradicals and me for arguing over grammar. Some (sad) people will restate a sentence in the passive voice to avoid using 'me' here, eg:
freeradicals and I were laughed at by them for arguing over grammar.
Your 'incorrect' example also changes the traditional position for the first-person pronoun (I/me): 'me and freeradicals' instead of 'freeradicals and me'. In practice, 'someone and me' occurs three times as often as 'me and someone', but 'someone and I' occurs thousands of times more frequently than 'I and someone'. If for whatever reason your sentence calls for putting yourself first, then choosing 'me' is always going to sound less awkward than 'I', even when you are the subject of the sentence.
The most important thing to remember though is that
there is no prescribed set rules of language. Language authorities - lexicographers, linguists and the like - write down the 'rules' of language to
describe it as it exists, not to
prescribe what is correct or incorrect usage. The history of the 'me and someone did something' form is much older than you are, and it is certainly not 'incorrect' in any meaningful way.