The road up to the park is filled with switchbacks. My tour ticket said that I should not take this tour if I was prone to motion sickness.
To illustrate: Our bus driver told this joke
A priest, a minister and a Haleakala bus driver approached the Pearly Gates. The priest was first and St. Peter told him to wait. The minister was second and St. Peter also told him to wait. The Haleakala bus driver's turn came, and St. Peter told him to go right in. The priest and minister protested. "We have given our whole lives to spreading the word of God and we are told to wait. Why does the Haleakala bus driver get to go right in?" St. Peter said, "Ah, but people often sleep during your sermons. On the Haleakala bus tour, they are all praying."
You need some sort of motorized vehicle to go to Haleakala so I recommend getting a rental car. There are also tour operators that offer bus rides to Haleakala for those who are scared of the drive. The road climbs quickly and is twisting and narrow at times. You will see numerous signs telling motorists to watch out for cyclists, which use this road to come down from the summit (and if you come early you'll also see the cyclists). There is no need for a 4-wheel drive in my opinion, any car can make the drive.
Although my mother (Beth) is a horse trainer, I am not at all interested in riding. I did see these horses at the top of the volcano