Are there any patterns on how to quickly determine meaning in a context or am I missing something really basic here?
In many situations, there is no quick way to determine meaning from context without having built up experience with the context in question. Unfortunately, it's simply a matter of knowing by doing.
But, in this case for instance, it simply wouldn't make sense that a romantic partner would redeem somebody in the sense of trading in a coupon for goods. So, logic should let you rule out certain senses of words. With any luck, they leave you with only a single remaining sense—which, by the process of elimination, is the only sense it can be.
In this case, without knowing anything else, I might narrow down the sense of redeem in the song lyrics to the following, in general:
2 : to free from what distresses or harms
3 : to change for the better : REFORM
4 : REPAIR, RESTORE
6 a : to atone for : EXPIATE
The other senses of the word don't seem logical. There are also multiple sub-senses of 2, which could make more or less sense.
But, in broad terms, "I want my love to redeem me" must therefore mean that they want their love to make them a better person in some way or rescue them from a bad situation.
What more specific sense of redeem is meant could only be learned from the rest of the lyrics. And if they don't provide enough context to narrow it down to exactly one sense, it probably isn't important enough anyway. (Many things, especially songs, are left intentionally open-ended.)