When the passion is read, the priest will read all three "parts" and there's really no indication as to who's saying what except that you're following along in your missal.

When the passion is sung, although the priest could do all three "parts" by himself, it's easier for 2 cantors to assist. The three parts are sung in three different keys (or octaves?) so even if only one or two are singing, the pitch will change as the dialog shifts from person to person.

Presuming three singers, the priest will sing the part of Our Lord with the cantors (or other clerics?) singing the "narrator" and "crowd". (I'm thinking the "crowd" is more than just that -- possibly more accurate to say "other".)

Perhaps Matthew will come along and fill in my question marks. He would know as he's one of our cantors.

If you're accustomed to hearing the Gospel repeated in English, don't hold your breath for the priest to go the extra mile and read through the Passion twice.  

Cathmom, If you have a traditional missal you'll be able to follow along with all (or most) of the Holy Week liturgy.  If you don't have a good missal, you can probably find the little Holy Week missal if you have a bookstore at your church or maybe your Church or church library has some extra copies.

I just checked Amazon and I see that you can get a St. Joseph Holy Week Missal published in 1957 for just $7.47 plus $3.99 shipping.  

I hope that you are in awe of your first traditional Catholic Holy Week!  I'm sure it'll be an experience to remember.