Yes you get interference - but a well constructed lens has introduced just enough phase shift in the wave front at every point that the interference is constructive. Because the lens is finite in size there will be some interference pattern observed at the focus - something known as "Airy's rings"
In fact - the way a lens works is precisely by creating a phase shift between light rays traveling along different paths, and after the phase shift the ray changes direction because that is the direction in which the interference is constructive.
The following diagram tries to explain this - I am using the conventional Huyghens construction to show that every point on a wave front can be considered a source of a wavefront that travels in all directions - with the final wave front continuing in the direction where all of these interfere constructively. The blue wedge is a prism - a very small piece of a lens. Inside the prism, the wavelength of the light is shorter (because of the refractive index of the lens), so the wave fronts (little circles) that represent a wavelength are closer together. You can think of a spherical lens as being made up of many prisms - each acting in the same manner (although the phase difference will change depending on the thickness of the lens). Note that in my drawing, the upper ray has exactly one wavelength inside the prism and two outside, while the lower ray has two wavelengths inside and only one outside. In both cases, the line connecting the wave fronts corresponds to exactly three wavelengths after the entrance plane of the prism. There are of course infinitely many rays between these two - if there were not, you would have something akin to a Young's Slits experiment setup, and would see interference patterns (several directions in which constructive interference can occur).
Incidentally - the picture you show in your question is very misleading. The rays don't "magically change direction" at the center of the lens - instead, they are refracted both at the entrance face and exit face of the lens. The following shows what I mean (in reality the angles are not quite as drawn - there is a thing called "spherical aberration" that is ignored here - but I hope you get the idea. I drew just the top few rays inside the lens in red; obviously the same thing is true for the bottom half):
UPDATE to explain how this works for a concave mirror:
If you take an arbitrary ray traveling parallel to the horizontal axis in this image:
You can compute its length as
$$length = D - y + \sqrt{h^2 + (f-y)^2}$$
Now if we want to set the length to a constant value regardless of $h$, we can say
$$\begin{align}\\
y + length - D &= \sqrt{h^2 + (f-y)^2}\\
y + C &= \sqrt{h^2 + (f-y)^2}\\
(y+C)^2 &= h^2 + (f-y)^2\\
y^2 + 2Cy + C^2 &= h^2 + f^2 -2fy + y^2\\
2(C+f)y +C^2 + f^2 &= h^2\\
y &= \frac{h^2}{2(C+f)}-C^2-f^2\\
\end{align}$$
Which describes $y$ as a parabolic function of $h$. In other words - in a parabolic (convex) mirror, the path length for all rays to the focal point is the same. So once again, there will be constructive interference at the focal point.
It is easier to think about this in reverse - i.e. what does it mean for light to be approaching from infinity? When we reference light from an object at infinity, we mean that the object is so far away that all light rays (strictly speaking in terms of geometric optics) from the object appear as if they are parallel and traveling in the same direction. When these parallel rays approach the convex lens, they get focused to the focal point of the lens.
In optics, the direction of light travel does not matter, thus when an object is placed at the focal point of the lens, the light rays will become such that they are perfectly parallel and traveling in the same direction. This is usually called "collimated" light.
Collimated Light
This method of collimating a light source (or object) is generally used in various imaging techniques such as shadowgraph and Schlieren.
Now what will happen if you place your hand in the optical path of the light? Well, theoretically, you should see nothing since no image is formed from parallel light. But in reality the light will never be perfectly collimated so you may see a very blurry version of whatever your object is. As soon as you place another focusing element in its path though, all those light rays will be focused at its focal point and you will (theoretically) have a perfectly formed image of your object.
Best Answer
Firstly, you must remember that the formula which predicts that the rays will meet at infinity assumes an ideal lens and environment- in real life the lens will not be perfect, so the rays will not emerge exactly parallel, and the environment in which the lens exists will absorb and scatter the light rays.
That said, saying the rays will meet at infinity means that they will not, for all practical purposes, meet unless another lens is placed in their path. If that lens is your eye, for example then the rays will be focussed to form an image on your retina.