Neutrinos are indeed tremendously penetrating precisely due to very little mass and no charge, many are constantly streaming through the entire world and you at any given moment. They are also way less likely to interact with matter than gamma rays. Since they very rarely do much of anything (you have to build ridiculous facilities to have a chance of detecting their uncommon interactions) they are not really considered as ionizing radiation.

Neutrons have much more mass but are still electrically neutral. Most materials have much more mass than a neutron so they bounce off and barely lose any energy in their collisions, sort of like when you throw a ping pong ball at a wall, which keeps them as highly penetrating in many materials. They only appreciably slow down when they collide with atoms similar in mass to them like the hydrogens in water, think like how a billiard ball will hit another and dump almost all of its energy into the similarly sized ball. So materials like water and concrete are good shielding for neutrons but lead not so much.