I suspect this is one of those language rules that requires a bit of practice; in fact I doubt there is any more effective way of learning it. What is common to both terms in that sentence is that I was stating a subjective opinion or belief about something. In the affirmative, I believe this will require practice and in the negative, I do not believe there is a good alternative to practice. When I suspect something, I have a suspicion about it - a belief. When I doubt something I have a sense it is not to be believed.
FumbleFingers example of "Don't suspect me" and "Don't doubt me" is brilliant and it illustrates why these "opposites" can be confusing. Suspect has a more complex use than doubt. The word suspect is a verb, a noun and an adjective - and in it's most common use it indicates something negative. A "Suspect" is a person suspected of committing a crime. We find someone's intentions to be "suspect". To say we suspect something is true shouldn't automatically indicate a negative tone, but it's related word "suspicious" almost always implies something negative - and that's not always the case. For instance, I could say, "I suspect you are smarter than you think", or "I have a suspicion you're going to be surprised by how high you score". Neither are negative beliefs. That said, I could just as easily say I had a "sneaking suspicion" you'll score higher than you think. That's sort of like a secret belief but there's a sinister tone to it.
The word Doubt is far more straightforward.