The HMs needed to complete HeartGold until Lance are Cut (to get through Ilex Forest), Surf (to reach Cianwood and Kanto), Waterfall (to reach Kanto), Strength (to get through Ice Path), and Whirlpool (to receive the Rising Badge in Dragon's Den).
I guess Surf is used most frequently since it's required to find the fifth gym leader, to get the Red Scale from Gyarados and infiltrate Team Rocket's base in Mahogany, and reach Kanto. I wouldn't count Fly since it's not really required to beat the game, it's more of a convenience. The other HMs (Cut, Waterfall, Strength, Whirlpool) are not used as much as Surf, unless you're exploring for items in caves or areas of interest.