Mormon doesn’t tell us how the Lamanites entered the land of Zarahemla, but the speed with which they arrived in the heart of the land suggests that they entered through the pass near Manti. We don’t hear about Manti after the book of Alma, so we cannot know if it was captured, or simply bypassed.
Coriantumr takes his army up the center of the land of Zarahemla. While he might have thought it would be the strongest defended area, it was, in fact, the least defended. The last major conflict had concentrated on both the western and eastern borders, and those cities had not only been physically fortified, but likely, at least, somewhat garrisoned. That left the center with fewer defenses.
Following the Sidon River valley allowed Coriantumr the most direct route to Bountiful, which, if captured, would significantly weaken the Nephite lands by dividing them in two without a good means of reinforcements and provisions crossing the center.