Should I mention at my PhD interview that I already got an offer from a first university? I have received a PhD offer from University A, but I'm also considering other programmes (say C and D), for which I will interview soon. 

I was wondering: should I mention during my interviews with C and D that I received an offer from A? What effect would this information have on my interview? Positive ("university A accepted him, so he should be a strong candidate"), or negative?

And let's say the answer is that I should not mention it, what if they directly ask me which other programmes I am applying to?
If A is your first choice, accept the offer and withdraw your other applications. If You would prefer to attend C, tell them (C) you have an offer from A but that you would rather attend C.