The only way to replenish Energy is through eating food that you either craft or purchase through Teng Fei.
Weather conditions such as Hot, Cold, and Mud can affect your Energy drain and speed.
Speed is determined for you and is designed to go at the fastest economical pace that will get your traveler to the next camp without collapsing. Holding the Shift key will make your traveler run faster. However, sprinting will cause your energy to drain more quickly.
Clothing, unlocked skills, current energy, and pack weight play a big part of your speed. In the top left of your screen, your maximum speed vs. weight is shown; there are speed pips that will be partially or fully covered up.
Check your "Buffs and Attributes" screen or hover over your Energy to see more detail about what is affecting your Energy drain.
Some foods can grant speed boosts, plan your trail snacks accordingly!