What is the correct order of nuclear decay modes for the change for the change from u-238 to u-234?

Answer is: α decay, β- decay, β- decay.

1) Nuclear reaction 1: ²³⁸U → ²³⁴Th + α (alpha particle).
Alpha particle is nucleus of a helium-4 atom, which is made of two protons and two neutrons.
Alpha decay is radioactive decay in which an atomic nucleus emits an alpha particle (helium nucleus) and transforms into an atom with an atomic number that is reduced by two and mass number that is reduced by four.
2) Nuclear reaction 2: ²³⁴Th → ²³⁴Pa + e⁻ + electron neutrino.
In beta minus decay, neutron is converted to a proton and an electron and an electron antineutrino and atomic number Z is increased by one. 
3) Nuclear reaction 3: ²³⁴Pa → ²³⁴U + e⁻ + electron neutrino