I am not an English-speaker, so I may be wrong in this, but I think being serious is an attitude a person takes, i.e., it may mean a person is determined to do what he/she is intended to do, or he/she is not taking things easy, while being careful is related to actions and how they are done.
I don't think you can distinguish them like that. You can definitely be serious in your actions. E.g.
I'm in serious training for the marathon. But, I must be careful not to get injured.
He committed a serious crime. And, after careful consideration, the court sentenced him to a term of imprisonment..
That's a serious mistake! You must be more careful.
There was a serious road accident. Caused by careless driving.
It's interesting that these words cause confusion. To a native speaker they seem very different. Although, both are for English typically flexible and can be used in a wide range of contexts. So, I guess there is no easy way to explain all the differences. But, perhaps the examples above help.