Grandparents who face parental resistance to their contact with grandchildren might consider requesting a mediation session with the children's parents. (In fact, some state courts won't consider your petition for visitation until the parties have attended mediation together.) Mediation means that you hire a neutral third party to help all of you create a legally binding agreement that everyone can respect and live with. For more about mediation and its advantages, see Nolo's article Mediation FAQ.
If mediation doesn't work and you end up in court petitioning for visitation rights, be prepared to testify about your relationship with your grandchildren, your relationship with the parents, any custody or visitation arrangements you had before the court action, and the last time you saw your grandchildren. Also, be prepared to discuss your personal history, including any medical troubles or problems with the law.
Remember that the court is scrutinizing you to decide whether you should spend time with a child -- and the judge won't hesitate to pry into personal matters. For example, if you're a smoker, you may be asked to refrain from smoking in the child's presence. If you don't, the parent may ask the court to discontinue your visitation. If you cannot, for health reasons, perform some tasks independently or correctly, your visitation may be supervised. This doesn't mean the court is prejudiced against you. You face standards similar to those faced by a noncustodial parent seeking visitation: The children's welfare is always given the highest priority, although the courts also give great deference to a parent's decision to limit visitation.
For a detailed discussion of child visitation, see Nolo's Essential Guide to Divorce, by Emily Doskow (Nolo).