At what speed do newly created protons move after a proton-proton collision? I read that when 2 protons travelling at near the speed of light collide with each other, they create 3 protons and 1 antiproton.
At what approx. speed those 3 new protons usually move once they are created after the collision?
 A: Proton proton collisions  belong to particle physics and are studied with the mathematical tools of quantum mechanics and special relativity which are very successful in describing experiments at the particle level and predicting new interactions.

I read that when 2 protons traveling at near the speed of light collide with each other, they create 3 protons and 1 antiproton.

This is misleading, two of the protons are the original ones, it is the proton antiproton pair that is created  in the interaction of the two original.
At the level of such large values of velocity one has to use special relativity, and has to define the inertial frame for which the velocity is high. Velocity is not invariant under the Lorentz transformations of special relativity, one uses the fourvectors and their algebra.

At what approx. speed those 3 new protons usually move once they are created after the collision?

Speed is not an invariant to Lorentz transformation.
One goes to the center of mass system, where the addition of the momenta of the two original protons is zero. That is how the LHC is designed, with two proton beams of the same energy and opposite momentum interacting, a center of  mass system.
One can do the energy balance and see how much,  at minimum, energy the two equal and opposite momentum protons should have  in order to be able to generate a proton and an antiproton. If all four particles , the two that were incoming and the two generated, have zero momentum each, the system energy from

should be $4m_pc^2$
This energy would have to be incoming by the two incoming protons and the minimum momentum in the center of mass needed for each of the two incoming protons can be calculated.
Certainly a higher value will be needed if one wants to detect the channel, because proton proton interactions go into many channels and different branching ratios apply.This  is due to  the quantum mechanical probabilistic behavior. Only the probability for a reaction occurring can be calculated, one can only study distributions of events. Single event variables  cannot be predicted .
