What to do with all those palms you received today?

Make a palm cross, roses or cones, a crown of thorns, or even a pretty cool looking bird.

From the Directory on Popular Piety and the Liturgy (139):

The faithful should be instructed as to the meaning of this celebration so that they might grasp its significance. ... Palms or olive branches should not be kept as amulets, or for therapeutic or magical reasons to dispel evil spirits or to prevent the damage these cause in the fields or in the homes, all of which can assume a certain superstitious guise. Palms and olive branches are kept in the home as a witness to faith in Jesus Christ, the messianic king, and in his Paschal Victory.

Finally, and above all, from today's Office of Readings (Saint Andrew of Crete): "Let us run to accompany him as he hastens toward his passion, and imitate those who met him then, not by covering his path with garment, olive branches, or palms, but by doing all we can to prostrate ourselves before him by being humble and by trying to live as he would wish ... Let us present the conqueror of death, not with mere branches of palms but with the real rewards of his victory. Let our souls take the place of the welcoming branches as we join in the children's holy song: Blessed is he who comes in the name of the Lord."