so are you saying that even though I could escape Earth's gravity going at sub-orbital velocities that I would still need to be going fast enough to avoid being pulled into the sun?
But no, it's not that you would be pulled into the sun. You would simply be in a solar orbit just like the earth, unless you continued accelerating and reached solar escape velocity (~600km/s). You can think of gravity like successively larger "wells" which you must escape from if you want to go anywhere. You can't just point a spaceship at a planet and go there.
Disclaimer: You can, you just need so much power it’s like flying with a helicopter from your bedroom to the kitchen, absolutely wasteful.
http://parkersolarprobe.jhuapl.edu/The-Mission/index.php 'Journey to the Sun.'
That's for escaping from the surface of the Sun. At Earth's distance from the Sun, the velocity needed to escape from the Sun's gravity is only 42 km/s.
What the GP said is that if when you reach Earth's escape velocity, there is still a long way to go until you have Sun's escape velocity (so you are on a solar orbit), and after you reach that, there is a huge distance until Milk-Way's escape velocity, and an even huger way until you escape the Local Group.
That's not what he said. Escape velocity is higher than orbital velocity (as ythn pointed out upthread). And escape velocity from Earth is still slower than Earth's orbital velocity around the Sun, which in turn is lower than escape velocity from the Sun's gravity at Earth's distance from the Sun.