Should I use Present Perfect or Past Simple in the first part of the sentence?
Your intuition that Present Perfect fits here is good; we typically use Present Perfect when a past action has continued relevance to the present, so since you are talking about how catching a cold results in staying home, it would make sense to use Present Perfect.
That said, if you did you the simple past, it would still be understood, but the connection between the two events would be weaker.
Should I use Present Continuous or Present Simple in the second part?
Out of those two options, you can only use Present Continuous. The simple present typically indicates habitual actions (e.g. "I stay home on Tuesdays"); it has some other uses but they wouldn't occur in this case. Present Continuous is used for actions in progress, which fits this case.
Alternatively, you could also use the simple Future:
I've caught a cold and I will be staying at home for a couple of days.
This could be interpreted slightly different from your original because the "staying" is now in the future rather than current, but in practice the difference is very slight.
Minor notes: At least in American English, in many cases, you don't need "at" with home. For example, you can stay home, go home, come home, leave home etc.
Similarly, at least in American English, you will see couple used without "of".