We can see stars that are many light years away from the earth but why can't we see a single satellite in the sky?
Satellites are very easily visible but only during specific times of the day. Those include just before the sun has risen and just before it sets depending on where you live.
To find a satellite go to a place where there is less city lights at the right time.
To actually get the time, you could download one of many apps (like the "ISS detector").
It's usually hard to imagine how big are stars. Most of the stars you see with naked eye are enormous giants much heavier and much brighter than our Sun. To compare sattelite with them is as sensible as to compare the powerful lamp with bacteria.
The only way to observe the sattelite is to see the sunlight it reflects. Usually the sattelite is a small object, often less than 1m in diameter, not optimized to reflect much light in your direction. Many of them are on the high orbits like geostationary orbit more than 35000 km above the sea level. You have no chance to see them. But even those on low orbit are more than 250 km above the sea level.
During day too much sunlight is scattered by atmosphere and it's impossible to see such bleak objects. During night the sattelites on low orbits above you will be in the Earth's shadow and hence reflect no light to observe. However there's very special time at late evening or early morning when the Sun is below horizon for you, but not the sattelites.
If you go outside city during those hours, if the sky is clear enough, the sattelites are actually very easy to see. Most of them will appear as rather bleak moving "stars". However some sattelites have parts that serve as a good mirror. If you see the reflection of the sun from such "mirror" you will see very bright flare in the sky, sometimes brighter than any star or planet may be.
The most famous are Iridium sattelites. You may find out when to look at the sky using for example this site. Be careful to correctly specify your location - even one km can mean significant difference in brightness. And remember, the number for brightness mean the apparent magnitude, so the lower it is the brighter the flare (i.e. -5 is a lot more brighter that -1, whereas +3 is very bleak).
Start with the ISS:
it can be very bright (-5.9)--second only to the moon (at night, duh). Moreover, its angular diameter is on the order of that of Jupiter: you can resolve it with modest binoculars.
If you can't wait for an ISS pass, go out tonight after sunset--you will see satellites in polar orbits (moving roughly north/south). This is a great time for infrared spy/military observations of thermal inertia--as night kicks in back on the ground. Many science platforms are in polar orbits, too. They are generally sun synchronous: they precess through the year so that they always cross the equator at the same local time (this removes unwanted diurnal variations in the data). A great example is NASA's A-Train.
If you see a satellite moving to the East--imagine where the Earth's shadow is, as you may see in the object enter it: it will start to fade and go to black.
Also: hunt for an Iridium Flash (or Flare), when an Iridium solar panel glints the setting sun (up there) back on you in darkness: they cast shadows.
Another resource is: