Lisa
In a 30 year old place with deteriorated low quality cabinets I recommend complete replacement. Given the time and cost of restoration, it may actually be competitive to install new. If that is not in the budget you cannot apply new coverings or veneers on an unstable (peeling) surface. A heat gun will probably be effective in helping you remove vinyl coverings. Chances are the vinyl is covering a hardboard or particleboard. You can buy veneers to cover face frames and purchase replacement doors and drawer faces.
Go price some new cabinets and see if it won't work for you. These cabinets don't appear to be worthy of your renovation effort and cost.