The U.S. bishops' Guidelines for the Reception of Communion, approved in 1996 and printed in many missalettes, welcome fellow Christians "as our brothers and sisters" and explain that members of churches not yet fully united with the Catholic Church are ordinarily not admitted to Holy Communion. Members of the Orthodox Churches, the Assyrian Church of the East and the Polish National Catholic Church are asked to respect the discipline of their own churches, and Catholic law does not object to their receiving. Other exceptions need the permission foreseen in canon 844 and the local bishop's directives. Everyone not receiving is encouraged to express in the heart a prayerful desire for unity with the Lord Jesus and with one another.
What to do during Mass is courtesy, not a rule; the Church's norms on posture are written for Catholics, not for guests. Stand and sit when the people around you do. When they kneel, you may kneel too, or simply sit quietly. At the sign of peace a handshake or a nod and "peace be with you" is all that is asked. At Communion, stay in your seat and let others pass. In some places people come forward with arms crossed for a blessing, but that is a local custom and not done everywhere, so if you are unsure, stay seated.
Good to know on a pilgrimage: Mass is usually celebrated every day, often in a side chapel of a shrine and sometimes in another language, and nobody checks who is there. Tell the group's leader or priest that you are not Catholic; it helps them look after you. You do not need to make the sign of the cross, say responses you do not believe, or pretend to know the liturgy to be a welcome guest.