If light is faster in vacuum medium than in air medium,
does it mean that we are seeing everything in a delayed manner since we live in air medium?
Is there any way to see things in actual speed i.e. in vacuum?
P.s. I'm not a physics grad, so I'm sorry if my question is trivial.
If you mean "do we see things in slow motion", the answer is "no". We see things with a slight delay, but at the same speed as if the medium was a vacuum.
The easiest way to see this is to think about what would happen over time. Let's assume we are looking at a clock, and the light from the clock gets to us slowly - say it takes a second longer than it would in a vacuum. Then when the second hand reaches "1 second past the hour", I see it at the top of the hour. But a second later, the information "it is now one second later" must reach me. Otherwise, all that information will end up piled up between the clock and me - and a person who just walks into the room would either see a different time than I see (they see the one second delay), or for them the situation would be different than it was for me when I walked into the room. Neither of those things make sense.
So - constant delay due to the extra time the signal takes; but other than that, no difference in speed with which observed events unfold.
As was pointed out by @hobbs, the actual difference in speed between light in vacuum and in air is tiny. With the refractive index of air at STP around 1.0003, the difference is not something you would normally notice. Light travels 1 meter in about 3 nano seconds; on that scale, an extra 0.03% adds about 1 pico second.
There is a delay, but you don't see something in slow motion.
Let's say a certain event happens between $t_0$ and $t_1$. If the medium between you (the observer) and the event is air, the light will indeed reach you with a delay. You will see the event beginning at $t_0+ \Delta t_{air}$ and ending at $t_1+ \Delta t_{air}$. So the timeframe of the event is not stretched, just uniformly delayed.
If there's a vacuum between the event and the observer, there is also a uniform delay. The observer sees the event begin at $t_0+ \Delta t_{vacuum}$ and end at $t_1+ \Delta t_{vacuum}$.
Because light travels faster in a vacuum than air: $$\Delta t_{vacuum} < \Delta t_{air} $$
So you see the event slightly earlier in a vacuum than in air, but the event lasts the same amount of time in both cases.
Now when do you see something in slow motion (or speed up)?
Let's use the same event in air, but change the situation a little bit. The event begins at $t_0$ at distance $d_0$ from the observer. The event ends at $t_1$ at a distance $d_1$ from the observer.
If $d_0 \lt d_1$ the begin of the of the event is seen by the observer at $t_0 + \Delta t_{air}$, nothing changes here. But for the end of the event an extra term needs to be taken in consinderation. Because the light needs to travel a longer distance $(\Delta d$), the end of the event is observed at $t_1 + \Delta t_{air} + \Delta t_d$. This means the event is observed later, but also the timeframe of the event is stretched out. You see the event in slow motion.
If $d_0 \gt d_1$. The way of thinking is the same, except $\Delta t_d$ will be negative. This means you see the again the event with the same delay, but you see it speed up because the light of $t_1$ needs to travel a smaller distance.
We never see anything in real time, if that's what you mean. The most common example in day-to-day life is the sun, which we see as it actually "appeared" eight minutes ago. Even moonlight takes just over a second to reach us. And when you read about supernovae being discovered, our telescopes are witnessing those events millions or even billions of years after they actually occurred.
But the delay for objects near us (say, a car across the street) is negligible to the point of being irrelevant; any delay added by the Earth's atmosphere slowing the light from that car is even more negligible and, for all intents and purposes, might as well not exist. The resulting illusion that we do see things in real time is what can make it quite unintuitive to think about time delays on the astronomical scale.
A much more obvious example of perceptual time delays is with sound; it's always fun trying to explain to demon-spawn for the first time that lightning and thunder actually "happened" at the same time.
The speed of light in any medium is given by the speed of light in vacuum divided by the relative refractive index of the given material. Now the relative refractive index of vacuum is obviously one. And the relative refractive index of air is 1.0003 at STP, and the value of C or the speed of light in vacuum is approximately 300000000 m/s ,so the difference between speed of light in vacuum in air and in vacuum =89973.0080976 m/s , therefore there is quite a lot of difference in the speed of light in air and in vacuum . But this difference does not bring a lot of difference in short distance observations ,so in case of our daily life observations it is not a matter of worry but for a long distance observation it might cause a problem.
The best metaphor here I think is to think of it as, we see everything on tape delay.
When they broadcast a sporting event, for example, maybe they delay the broadcast 15 seconds so they can have a chance to cut to commercial if something illegal to broadcast happens. Or in the case of the Olympics maybe they delay it even some number of hours so to show it at a more convenient time for the viewer.
That means that you see things (on tape delay) that occurred some time before, right? But you still see them happen at the same speed that they occurred originally; the football player still runs at the same apparent velocity, the ball still is kicked with the same power and apparent velocity. None of that changes. The only difference is that it occurred fifteen seconds before you saw it.
The only reason this would be not true is if some of the light you see were to travel through a medium of air and some of it were to not travel through a medium of air. Then your vision would be distorted (only very very slightly, but still). This would be similar to seeing underwater (or seeing things that are underwater) - the water distorts the light some, not only slows it down but also diffracts it some, in a different way than the air does; thus you have a visible difference. But since everything we 'see' comes through the same air (unless you're looking at something through a space-based telescope I suppose?) there is no apparent distortion since it is all slowed down at the same rate.
Light travels about one foot per nanosecond. One nanosecond is the period of anything vibrating at 1GHz. If you're in a typical room in a house or office building, you're looking at things maybe ten, twenty, orthirty feet away. What you see is how things were 10, 20 or 30 nanoseconds ago. For the everyday things in an typical Human's life, this is so small it hardly matters. Only physicists and radio engineers care.
What about the delay due to air? Vacuum is by definition a "medium" with an index of refraction 1.00000 while air, at the surface of Earth, at a comfortable temperaturn and normal pressure and density, aka "STP", has an index of refraction about 1.00029, for visible light.
Out of a travel time or 10, 20 or 30 nanoseconds, the slowdown caused by light is a fraction 0.00029 of that, which is 0.0029, 0.0058 or 0.0087 nanoseconds.
What about when you're outdoors looking at mountains ten miles or so away? The regular speed of light means a delay of about 52000 nanoseconds, or 52 microseconds. The effect of air, compared to vacuum, amounts to 0.00029 of that, about 15 nanoseconds.
The exact index of refraction of air depends on temerature, density and humidity. The NIST has a page detailing this variation.
It should be noted that this is only a time delay. Any repeating activity will appear to occur at the same speed. If a light is blinking every 1.000000000 seconds, you'll see each blink several nanoseconds later, but still ocurring every 1.000000000 seconds.
When looking at the Moon, Jupiter, stars, or anything outside Earth's atmosphere, you're looking through vacuum mostly, but there are several miles' worth of air between you on the ground and empty space. Our atmosphere tapers off gradually. If you brought down all the thin air in the upper fringes and packed it down so it's all at STP, I think it comes outto be about 10 to 15 miles thick. Well, that's about the same as the mountain example I just gave. So you would see a supernova go off about 15 to 20 nanoseconds later due to Earth's atmosphere, compared to if the air weren't there.
Fun fact: radio astronomers can measure the delay of microwave signals from pulsars due to the interstellar medium. This medium is what most normal humans call "vacuum" but it's not perfectly vacuum. The index of refraction is tiny, and depends strongly on frequency. Measurement of this variation tells astrophysicists something about the few neutral atoms, H2 molecules, and free electrons and protons, drifting around between the stars. It's mostly hydrogen molecules, at about one million per cubic cm.
