We do not know if the universe is expanding. IF it is, then it is expanding a LOT faster than that.
We do know that SPACE is expanding: between any two bits of space, more space is "growing". The local rate is extremely small: space expands everywhere, including inside every atom. Over a distance of 2 metres (the height of a tall human), it will expand by the size of one atom over a 70-year lifetime.
Since this happens everywhere, it adds up over long distances. For example, over a distance of 1 million parsecs (= 3.26 million light-years) the rate of new space being added is equivalent to adding 68 km per second (actually closer to 69.3 -- I still use 70 for rough estimates).
A million parsec is a long distance. When you leave Earth and go to (let's say) Mars, this is comparatively a very small distance (0.0000081 parsec).
Over that distance, the rate of expansion of space is... 0.000000000566 km/s
equivalent to almost 2 cm per year.
So yes, in a way, if you were able to remain "dead in space" relative to Earth, at a distance of 0.6 astronomical units (Mars's closest approach, roughly), the expansion of space would cause the Earth to "run away" from you at a rate of 2 cm (less than one inch) over a whole year.
Throw a bottle out of a vehicle moving at 2 cm per year, and a police officer on foot would be able to pick it up and write you a ticket for littering... all that without taking a step.
The Local Group of galaxies (the group we belong to) is just about large enough to have a distance of a million parsec somewhere. For example, the distance between our Galaxy and the Andromeda galaxy is a bit less than a million parsecs; the expansion of space is equivalent to approximately 40 km/s. However, as both galaxies are "in orbit" around the Local Group's barycentre, we are moving towards each other at more than that speed, therefore expansion will not be sufficient (over such a "short" distance). to prevent the collision.
The other large spiral galaxy, within the Local Group, is Triangulum (the name of the constellation in which it is seen). That galaxy is almost a million parsecs away (0.9 Mpc).
The Local Group itself is "falling" towards the centre of mass of the Virgo supercluster (a collection of clusters and groups of galaxies) and it carries everything in the Group, including us and the other 40-some galaxies that form the Local Group. Our Galaxy is moving at a speed of 400 km/s relative to the Cosmic Microwave Background radiation (the closest thing we have to a rest reference). The distance to the Virgo supercluster's centre is "only" 20 million parsecs (20 Mpc). Over that distance, the rate of expansion is (roughly) 20 * 70 = 1,400 km/s (less than 5% the speed of light).
Working backwards, if you calculate the rate over increasing distances, until the rate of expansion equals the speed of light, you find a distance of 4,225 million parsecs (13,772,000,000 light-years).
That is why we say the Observable Universe is 13.8 billion light years in radius.
The whole universe? All we know is that the whole universe is bigger than that. We cannot see anything beyond that distance because the rate of expansion, over such a distance, is equal to the speed of any information (light) trying to come our way.
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On a separate issue, Earth is orbiting the Sun at a speed of 30 km/s. All of us, on Earth, are sharing in this speed (relative to the Sun). If you leave Earth in a rocket, you will be moving into space at a rate of 11 km/s (escape speed) relative to Earth. However, you still have this orbiting speed of 30 km/s relative to the Sun.
If you leave Earth at a speed less than 11 km/s (but more than 8) you will remain in orbit around Earth and it will never "run away" from you. When in orbit around Earth, you are MUCH closer than Mars and, therefore, the rate of expansion of space is a lot less than 2 cm a year.
If you leave Earth with a speed greater than 11.2 km/s, you are no longer in a closed orbit around Earth. However, you are still in orbit around the Sun, because the escape speed, from the Sun, at our distance, is a bit more than 42 km/s.
A few probes have reached solar escape speed. Some of them have left back in the 1960s. Although they will never fall back towards the Sun (they are going faster than escape speed), they are still inside the gravitational sphere of the Sun.
So, even if you manage to get your rocket to go at the same speed as these probes did, you would still need hundreds of years to leave the Solar system.
However, because you would have reached "escape speed" (relative to the Sun and relative to Earth) then you could truly say that the entire Solar system is "flying away" from you (or you are flying away from it - same thing, because space does not care what you measure from) and it would be impossible for you to ever "catch up" with Earth... unless you turn the rocket around and kill your speed by accelerating back towards Earth.