I bought the seeds from the Realseed company, and put them into a heated propagator. But, to be honest, I had them in the greenhouse for about a week before I asked for advice on the forum, and once I put them into the propagator they germinated in a day.
I have never tried growing melons before, this one is watermelon, and my other seeds are for a little melon, can't remember the name, but I am going to sow them at the end of April. I will grow them in the greenhouse, than goodness I have got an internal waterbut in my greenhouse, otherwise I would be having a nightmare trying to get enough water to the watermelons, they drink a lot I would imagine.
Don't know about using seeds from a bought melon, perhaps someone else on the forum might be able to advise you.