What is good to relieve the persistent cough with and without mucus?
Something we can get over the counter or what we can find at home.

A few thoughts...

First, do you have an infection of some kind? The signals for that are: fever, aches or chills, nausea, the usual. If you do have an infection, then you might want to see a doctor. They will be able to try tell if it is viral or bacterial, and if it is bacterial, they can give you antibiotics (I know you want over the counter remedies, but I am just including this for completeness.) Also, if you do have an infection, you don't want to just stop the coughing. The coughing is clearing gunk out of your lungs, and is part of the way your body fights off the illness. (The same is true of a runny nose.) You can make the cough more productive by taking guaifenesin syrup or tablets. Even generic comes in both formats.  Guaifenesin is an expectorant, i.e., it helps you bring up mucus. You can find it in generic syrup, or in Robitussin.  If you buy Robitussin, make sure you get the variety that has nothing but guafenesin. Various types also have decongestants, cough suppressants, and so on. There is a tablet form called Mucinex which is longer lasting, but more expensive. You should take it with a full glass of water. Other home remedies which have a similar effect include chicken soup and foods with lots of red pepper. They increase fluid secretion so your cough becomes more productive. (Some people recommend a concoction made with honey, lemon and red pepper. But spicy salsa probably works as well.) But if you don't have an infection, or you may have allergies, in which case the cough isn't useful to your body. In that case, you can take a cough suppressant like dextromethorphan (found in certain types of Robitussin and other drugs.) However, dextromethorphan in particular suppresses coughs by acting on the brain, and not surprisingly, it has a variety of possible side effects, like sweating, itching, nausea, dizziness, etc. So another possibility is theobromine, which is found in chocolate, especially dark chocolate. 50 grams of dark chocolate, or about half a 3 oz. bar of chocolate, should have enough to have an effect, though you can take more. (Ten teaspoons of cocoa powder would have about the same amount of theobromine.) You can of course try cough drops. They are cheap and generally harmless. I try to get sugarless ones to help protect my teeth and gums. Try to avoid ones that have benzocaine, which is a topical anesthetic, and possibly irritating, though it can reduce throat pain temporarily. Good luck!

Sources: http://en.wikipedia.org/wiki/Guaifenesin, http://en.wikipedia.org/wiki/Dextromethorphan, http://en.wikipedia.org/wiki/Antitussive, http://en.wikipedia.org/wiki/Theobromine   The intent of this post is for hard times when you can't get to a doctor when  you think that is the best course of action. In such a hard time situation,  remember rest, warmth, and hydration is always a good fall-back. And if vomiting is involved, you can try use pet odor spray, too, if you have it.   I remember using these ideas once when deployed in Korea during the winter. All my buddy and I and my friends could do was check on him as he lay in his sleeping bag on a cot in a tent, and he healed over a day or so. During this time all we also could do was make sure he was warm and hydrated, and let him rest and sleep. Thank goodness he was not vomiting, but he was still a sick person. Now he had seen a doctor, who pretty much recommended the same thing. And sure enough he got better and returned to work.