Gravity is an extremely weak force. Note how easy it is for you to throw a ball upwards - counteracting the pull of the whole planet with scarcely any effort. While the escape velocity of the Sun's "surface" is very rough with respect to the spaceships we have, it's tiny compared to the speed of light.
Light always travels at the speed of light - almost all of the visible light from the Sun that reaches us has been created in the photosphere, not in the core (where fusion occurs). Even the highest energy photons are absorbed or scattered almost immediately in the hot dense plasma. This is a good thing for us - the photons created in the core are all well in the "gamma range", while the ones radiated from the photosphere are mostly visible and infrared.
The "it takes thousands of years for light to escape our sun" is a bit misleading, and certainly confusing if you don't have a decent quantum physics footing. It doesn't literally talk about a little ball being produced and then travelling at the speed of light for thousands of years before it travels the distance of less than a light second. It just takes a lot of time for the energy to propagate through the bulk of the Sun (two main processes are important here - absorption + radiation, and convection), in some ways similar to how it takes a lot of time for Earth's heat to propagate all the way to the crust - and yet you wouldn't say that the (infrared) light took "thousands of years to get from the core to the crust". It didn't - it just wouldn't have been emitted if the heat wasn't released that long ago in the first place.
The deal with neutrinos is that they have very little interaction with anything even at the large densities of the Sun's core (about 150 times the density of water under standard atmospheric pressure on Earth); only a tiny fraction of the neutrinos produced is actually absorbed on their way out of the Sun, so they do actually travel the distance in "a straight line" - and with their speed very close to the speed of light, it's a very short time indeed. That's why if the Sun magically stopped all fusion right now, we would know in just a few minutes, even though the change in the Sun's energy output would be very small for hundreds of years to come (though as far as I'm aware, noöne has done a full-scale simulation to see how exactly that kind of thing would proceed). Most of the interactions you work with daily are the result of the electro-magnetic force - things like touch, sight, smell... neutrinos don't interact electro-magnetically at all - they only interact gravitationally (remember how we said that gravity is a very weak force?) and "weakly" (about five orders of magnitude weaker than the electro-magnetic force). And the weak interaction is very limited in distance (unlike both electro-magnetism and gravity).