During an oceanic crossing, both pilots monitor the radios. Position calls are made at specific intervals, including weather and fuel reports. Plotting charts are kept which record position, altitude, and time, and which help ensure that accuracy is maintained (and is shown on the chart to have been maintained).
I like to read, albeit brief bits of reading in between monitoring or operating. Just lately I have a Kindle, which enables quick download of news magazines, or books, and fits neatly in a flight case. Reading isn't done in terminal areas or during times of high radio traffic or other busy activities, but on some legs when it can be 45 minutes between position reports, reading is good, safe, and keeps one awake and sane.
Company position reports, fuel calculations, equal time point calculations or recalculations, and other activities take place enroute.
When operating in uncontrolled terminal areas, the specific procedures depend on the company. I can't speak for flybe, but we have very specific limits as to when and where we can operate under visual flight rules, and the distance from the airfield from which we must be under positive control and on an instrument clearance when arriving or departing. We have limits on when we can cancel our instrument clearance.
Various tools supplement the single most important safety device, which is good old fashioned Mark 1 Eyeballs, when it comes to looking for traffic (other aircraft). The Traffic Collision Avoidance System is the most obvious, and this system interacts with transponder devices in other airplanes to help work out conflict solutions...it provides directions in real time to prevent airplanes from smacking into each other.
My "day job" airplane is too big, and requires too much runway to operate into small, uncontrolled airfields. None the less, we do operate regularly into less-than-ideal locations around the world. Such environments require more of a crew, because the onus to avoid terrain, other aircraft, and sometimes unusual hazards are more fully on the flight crew and not air traffic control.