# Put the harness on the horses; mount up on the horses

A "harness" is a set of straps placed on a horse so that it can pull a chariot. Here "mount up on the horses" means that the soldiers rode in the chariots that the horses were pulling.

# helmets

A helmet is armor which protects the head in battle.

# polish the spears

This phrase means to "sharpen" the spear.