Shrine is the word with pilgrimage built into it. Canon 1230 of the Code of Canon Law defines a shrine as a church or other sacred place to which many of the faithful make pilgrimage "for a special reason of piety, with the approval of the local ordinary." The reason may be an apparition, a saint's tomb, a relic or a place in the life of Christ. A diocesan shrine is approved by the local bishop, a national shrine by the bishops' conference, which is why the Basilica of the National Shrine of the Immaculate Conception in Washington, D.C., carries that name, and an international shrine by the Holy See. Sanctuary (santuario, sanctuaire) is simply the local word for shrine, so the Sanctuary of Lourdes and the Shrine of Lourdes are the same place.
Basilica is a title of honor and says nothing about pilgrimage on its own. Rome's four papal basilicas, St. Peter's, St. John Lateran, St. Mary Major and St. Paul Outside the Walls, are the major basilicas; every other basilica is a minor basilica. Under the Holy See's 1989 norms the Pope grants the title to a church of particular importance for liturgical and pastoral life, usually one with some renown such as a saint's relics or a venerated image. A minor basilica may display the papal crossed keys.
Cathedral is a matter of office, not of size or beauty: it is the church of the bishop of a diocese. Rome's cathedral is St. John Lateran, not St. Peter's. The cathedral of Santiago de Compostela is a cathedral and a shrine at once.
On an itinerary the words tell you what to expect. Canon 1234 asks shrines to supply "the means of salvation" more abundantly, especially Mass and confession, which is why the large shrines keep confessors in many languages. None of the three words tells you about opening hours or tickets.