I was just wondering as I passed the airport this morning why, whilst we are flying, do we never see any other planes nearby? There must be millions of planes and millions of people flying each day, I appreciate the sky is massive but these planes must follow roughly the same routes so how is it that they never seem to pass by any other aircraft?
If you're not seeing them, you're not looking.
I see planes all the time when flying commercial. Obviously, I see them a lot more when close to a major center, and a lot less when in the sparse parts of the Pacific Northwest (Montana, Idaho, Oregon, Washington).
I think spotting them may take some practice. Visually, a plane 10+ miles away is a very small dot, and easily missed. Try looking for contrails and be sure to look at altitudes above and below you. Scan the sky in narrow, 10° concentrated "fields of view" to scan from left-to-right all the way across the sky.
Sometimes you have two for the price of one:
From my personal experience, seeing other aircraft is really common, as already commented and answered, even far from busy airports. The image above is taken over Germany, and the other aircraft is just a bit higher:
Of you look at site like Flightradar24.com, you'll see aircraft following the same route, and close enough to allow each other being seen by passengers. Screenshot (look at the scale):
Aircraft are usually routed one in front of the other, often at different flight levels.
Situations where another aircraft would be flying alongside yours, where they would be easily visible, are fairly rare. They do happen though. I was flying from Chicago to Boston last Thanksgiving and there was another heavy alongside about 2000 feet below and 3000 feet to port.
If aircraft are on an intersecting airway, they will generally be several miles away, so they will be very small, and will usually be at different flight level.
That's not true, you DO see other planes in the sky. I fly very infrequently (on average probably 1 flight a year) but I've seen other planes in flight. I remember on one flight, my airplane entered a quite steep bank to the left, I was sitting by the right side window, and I saw another passenger jet fly past above me, it was rather close, probably a 1000ft altitude difference.