If since purchasing my home I got married but my wife's name is not on the home. and I were to die, would the home naturally go to her providing she could continue the payments?

Your home would not necessarily go to your wife.  If you have no will, the home would pass according to your state's law.  This is usually listed in the statute of descent and distribution, but some states, like Florida, have constitutional provisions for "homesteads."  In most states, the home would go to your wife.  However, some portion of it may go to your children, if you have children.

The best thing to do is consult a life or estate planning lawyer and prepare a comprehensive life plan.  This would be tailored to your specific circumstances.  The home could be passed to your wife by preparing a new deed, a will (but this would require a probating your estate), or a trust.  A comprehensive life plan would also make provisions in case you or your wife became disabled, if you and your wife both passed away, and other circumstances.

A comprehensive plan does not have to cost a lot of money.  If you have simple circumstances and clear wishes, it can be prepared quite economically.  In any event, your money is better spent in planning than in cleaning things up later.