We have had 11 men from our parish ordained in the last 10 years, so I’m pretty familiar with the process.
Generally, there is a Mass of Thanksgiving, which the priest celebrates. Generally, he is not the homilist, his pastor will be.
The recent ordinand will often have friends from the seminary act as servers and Readers, and any who are instituted Acolytes will fulfill any needed EMHC roles.
It is customary for the recent ordinand to receive a blessing from his parents, and to offer one to them in return. He will then give his mother a particular memento from his Ordination. After his ordination, he was given a linen cloth to wipe off any excess chrism from the ordination itself. This is traditionally give to the priest’s mother ( and is generally buried with her, if that is her wish).
Afterward, at our parish, will be reception of coffee, donuts and light snacks. During this time the new ordinand will offer blessings at a prie-dieu.