I know so far that escape velocity on the surface of Earth is 11.2 km/s Wikipedia says that escape velocity is the minimum speed needed for an object to escape from the gravitational influence of a massive body. Isn't it the initial speed, which should be equal or overcome Ve in order to escape the gravitational pull? Rockets or other spaceships use fuel in order to propel out of the planet, but they are constantly burning fuel and accelerating. As far as I know rockets do not get launched at a speed which equals or overcomes escape velocity. Do they later have to achieve it in order to escape? Or can they just constantly be accelerating but at a speed slower than Ve?
The escape velocity is a bit of abstraction, an idealized situation. It applies, to a degree, but about never as the exact value of velocity of any object.
First off, a rocket launched at that speed from the surface would burn up in the atmosphere long before reaching space - but in practice the segment where an interplanetary rocket accelerates is short enough comparing to its total trajectory, you can approximate the "initial speed" with "speed at the end of the escape burn"; it's near enough and short enough the error is rather small.
Next, a rocket given exactly the escape speed will escape Earth's gravity and... fly nowhere interesting. Just orbiting the Sun on an elliptic trajectory reaching maybe halfway towards Mars or Venus (depending on direction of escape). You want to give the rocket enough of initial velocity to reach the actual target, not just escape (due to Oberth effect, it's best done early on, combining escape and transfer burn. It's massively wasteful to do this only after the rocket escaped Earth's gravity.) So the rocket at the end of the burn will possibly move slightly faster than Ve.
Next - we're not alone in empty space. The escape velocity is usually an abstraction for a two-body system, rocket and planet, and nothing else. No gravitational field of the Moon. No Sun gravity and its lagrangian points. So, the actual velocity will account for these.
Next - issues of low-thrust transfers. If the probe uses a low-thrust engine, like the ion engine, the burn duration will no longer be negligibly short. Loss of velocity due to climbing out of gravity well, vs gain due to thrust; the probe never achieves escape velocity, it just "climbs out" of the gravity well, instead of being ejected.
And then delta-V, Oberth effect and gravity losses. If your rocket packs 11.2 km/s of delta-V in its tanks, you're getting somewhere halfway to the Moon, or thereabouts. Nowhere near escaping. Atmospheric drag, gravitational losses, loss of velocity due to climbing to LEO from surface altitude, that all needs to be accounted for.
The escape velocity is nice as a "milestone", a marker to compare requirements and actual velocities against, and to train/play/learn with simulated, idealized models, but it never applies directly - it's just an arbitrary, if evocative data point to compare actual velocities to.