Why is the amu of chlorine-35 less than 35? My book says that a proton weighs 1.0073u, a neutron weighs 1.0087u, and an electron weighs 0.00055u.

Now, why is the mass of chlorine-35 equal to 34.969? Are there not 17 protons, 18 neutrons, and 17 electrons? I calculated it and it sums to around 35.29. Where did I go wrong?
You need to account for the energy released when nucleons and electrons come together and form a Cl-35 atom. Its called the Binding Energy.

This sort of equation can help to explain :

(Rest Mass Energy of Individual nucleons,electrons*) - (Various Binding Energies) = (Rest Mass Energy of Natural)

*edit