If you get sick after getting the flu shot, it's unlikely to be the actual flu - colds are very different and usually non-harmful infections for most of us. If you do get the flu full force very shortly afterwards, it's either a strain that wasn't included in the shot (which sometimes happens, scientists make their best educated guesses but they are only human), or you were infected and sick before you got the shot (which then doesn't help you out - not the shot's fault!). The very minor symptoms of being slightly feverish and feeling lethargic are normal, not the actual flu, but your immune system recognizing and destroying the viruses.
If you get the flu later on, you will experience reduced symptoms and have less risks of developing very serious complications. Your body will only need a couple of days to fight off the infection versus the 2 weeks it would take without being vaccinated. This is what helps us survive through an infection and prevents the really bad symptoms that level us. The flu shot does NOT prevent the flu from happening in your body, it just makes it a whole lot less bad.
My dad believes the myths that 'getting shots makes your immune system lazy' (very much the opposite is true) and that 'he got the flu because of the shot'', all which is false. I worry as he's a T2D himself and now doesn't get the shots anymore. The chances of getting the flu and suffering severely are MUCH higher than any of the minor risks associated with vaccinations, so in my eyes it's foolish not to get it. I get them every year and I haven't been seriously ill since. Yes I've had colds, fevers and the usual seasonal woes, but nothing that made me miserable or close to death. It's worth it.