Airlines fly thousands of flights every day and it's been "statistically" shown that taking a flight is FAR safer than driving to work...
The requirements for becoming an airline pilot are some of the most demanding you'll find. Every 12 months airline pilots have to pass a very thorough medical exam (private pilots are required to take a less stringent medical exam every two years...) These exams are given only by FAA certified doctors (not general practitioners)
"Recurrency training" is something pilots go through on a regular basis and involves hours of classroom and flight simulator time where emergency procedures are practiced for virtually any problem that might come up.
Crew Resource Management has become a major part of pilot training that keeps pilots working together as a "team" while they are acting as pilots.
Air Traffic Control Centers are radar facilities that provide "separation" while flying - all airline flights are conducted as IFR flights (Instrument Flight Rules) even when the skies are clear and visibility is unlimited...
Pilot Reports are made between Pilots and ATC on a regular basis, making sure everyone is where they're supposed to be (altitude, heading and location) as well as weather conditions and any other information that the pilots need.
Before every flight every plane is thoroughly inspected by maintainance (which can take 4-8 hours) and the pilots do a thorough "preflight" inspection as part of their regular checklist...
While things can (and do) go wrong - "statistically" speaking, accidents are far and few between...