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Right now, in the center of the Sun, there's a nuclear reaction converting hydrogen to helium, producing neutrinos, photons, & positrons. A neutrino from this reaction will pass through you just over eight minutes from now. The photons won't arrive for at least 100,000 years.
It's not that neutrinos are inherently faster than light. It's that they can pass through the Sun -- and everything else -- without interacting, but gamma rays can't. It takes a photon over 100,000 years to fight its way out of the Sun, and 8.3 minutes to reach Earth from there.
Neutrinos don't travel through the vacuum of space *quite* as fast as light does (nothing with any amount of mass can), but it's close. When a supernova occurs in another galaxy, we receive a neutrino burst before the light, because the light got trapped but the neutrinos didn't.
So far, we've only managed that detection once (with Supernova 1987a and the Kamiokande neutrino detector), but next time the right kind of supernova happens close enough to Earth, we should see it again. aavso.org/vsots_sn1987a
There is something called the SuperNova Early Warning System that astronomers can sign up to in order to receive an alert of a neutrino burst, to give us time to point our telescopes in the right direction. It hasn't been triggered yet, but we wait in hope snews.bnl.gov
By the way, that neutrino from the core of the Sun that just passed through you wasn’t alone. There are 100 BILLION of them zooming through just your thumbnail EVERY SECOND
You won’t feel it, though. Chances are, not a single one of those neutrinos will interact at all with a particle in your body over the course of your entire life.
