There is no real record or account of it's location and travels after the Last Supper. Most likely, it simply remained in the household of the people Jesus and the disciples dined with. For all we know it no longer exists.
Luke 22
8Jesus sent Peter and John, saying, "Go and make preparations for us to eat the Passover." 9"Where do you want us to prepare for it?" they asked. 10He replied, "As you enter the city, a man carrying a jar of water will meet you. Follow him to the house that he enters, 11and say to the owner of the house, 'The Teacher asks: Where is the guest room, where I may eat the Passover with my disciples?' 12He will show you a large upper room, all furnished. Make preparations there." 13They left and found things just as Jesus had told them. So they prepared the Passover.
Of course, there is also a supposed grail from the Crucifixtion. But there is no mention of it in the scriptures, I'm prettys sure it is atradition of the Orthodox Chuch, because they drink both wine and water at Holy Communion. The theory is Joseph of Arimathea took possesion of this cup when he asked Pilate to take the body of Christ to his tomb the tomb.
John 19
34Instead, one of the soldiers pierced Jesus' side with a spear, bringing a sudden flow of blood and water.
Neither one probably exists anymore, or if one does it's just some cup floating around somewhere. It bears no meaning in the church, it is simply a relic, finding it would mean nothing because it would still just be a cup.