I know that aircraft are pretty well built and maintained, but I am certain that while flying these aircraft also suffer from some or the other mechanical issues. I believe redundancy plays a huge part in keeping these aircraft in air, but how many times are the passengers really oblivious to the problems? What are some of the most common problems that pilots are used to?
I don't think you'll find a definitive answer because most will be just considered routine items, but if you're including non-safety-critical items then I imagine the answer is "lots". Passenger aircraft are huge, complex, mechanical devices with many many parts. It is accepted and expected that some of these things will occasionally fail.
Most of these can fail gracefully without impacting the flight in any way beyond a note in the tech log. Not only that, but lots of things can be known to have failed but still allow the flight to go ahead as normal.
You may wish to explore the idea of Minimum Equipment Lists. These provide exhaustive lists of items which can fail and flights can still be dispatched as normal. Note this isn't the same as an in-flight failure occurring and continuing to the destination - items in these lists can be failed before you even take off.
For example, here's one for the Airbus A320: https://fsims.faa.gov/wdocs/mmel/a-320%20r21.pdf
Everything listed in that document can be failed in some way, but a flight still operate as normal (with exceptions, as per the remarks column)
For example, and to highlight the level of detail, an Airbus A380 does not need to be grounded because the "PA In Use" notification light doesn't work.
It used to be normal for the airliners of the day to have multiple non-critical fault alarms active at takeoff. Airlines tended to take the view that if an alarm is non-critical then there is no problem, no need to hold the aircraft back. Pilots complained that there were often so many that spotting a critical warning among them could become a problem. Pilots also often disabled them (especially audible bleepers, buzzers and such) to reduce the level of distractions. Keeping the level to manageable proportions is an important part of modern flight deck instrumentation design; the fault indicattions are still there, but not intrusive. Different airlines tolerate different levels of fault indication.
I was once a passenger in a venerable Dakota, still being operated by a commercial airline. Several rows of rivets were missing from the trailing edge of one elevator. Most had been replaced by twisted wire, but some holes were just left. It had evidently passed inspection nonetheless. Most passengers never noticed, and the trip was uneventful.
The general approach to passenger information is often, as they say, "what the eye don't see, the heart don't grieve over".