You need to think about how hard something is being pushed, not how much speed is being added. Unless an engine is pushing a rocket hard enough, which is called its thrust, it can't overcome the force that is pulling on the rocket, gravity. Also, the bigger the whole rocket is compared to the the engine, the more thrust is needed to move it. So, engineers talk about the thrust-to-weight ratio of a rocket, and normally write that T/W. To be enough to move the rocket upward, the T/W needs to be larger than 1.
The NEXT ion engine has a thrust of 236 milliNewtons, and a Newton is the force it takes to accelerate 1 kg by 1 meter per second in 1 second. Since an ion engine weighs much more than 236 grams, it can't lift even itself into space. The Space Shuttle solid rocket boosters have a thrust of 12 million Newtons each.
The reason why ion engines work so well in space, is that when they are turned on, the spacecraft is already moving in orbit, so it can't fall to the ground any more, and it is out of the atmosphere, so there is no friction slowing it down. In that situation, even though the engine is only thrusting a tiny amount, nothing is overpowering it any more, so it slowly adds up and the space craft accelerates. Because ion engines are extremely efficient with fuel, and it takes years for a deep space probe to go places so there is plenty of time for the acceleration to add up, they are very good for ships that have to go very far.