Let's get this out of the way up front: The helicopter controls in this game are not intuitive. They could have been implemented better and should be patched. That said, I think I have a decent amount of insight into the way they function that will hopefully provide some context and make them easier to get used to.
I'm a helicopter pilot by trade, and I can tell you from experience that the way a helicopter actually handles doesn't translate very well into games. You're constantly manipulating the cyclic (think the stick in a fixed wing aircraft), collective (the up/down lever, simply put), throttle (to a very limited extent), and pedals (heading control) throughout the flight.
A helicopter has two phases of flight, defined by a concept called effective translational lift. The simple version is this: When a helicopter is hovering or flying at very low airspeeds (20kts or so and below), the downwash from the rotor blades is being re-circulated through the rotor disk, resulting in inefficiency. In this phase of flight, the aircraft can fly sideways, climb/descend vertically, rotate about its axis, etc. When the aircraft accelerates, it eventually outflies that downwash, meaning that only clean air is circulating through the rotor disk. When it passes through effective translational lift (ETL), it essentially flies like a fixed wing aircraft (minus stall considerations and other small factors irrelevant to the game).
To put it in context that makes sense within the game, think of it this way. When you initially takeoff or when you aren't holding shift (sorry, I'm not familiar with console controls), you're effectively in hover mode. You're below ETL and it's flying like a helicopter (minus the inability to strafe. That's bullshit). When you hold shift and accelerate, watch for the nose to pitch up slightly. That's your indication that you've passed through ETL. At this point, as long as you hold shift, it'll fly just like a fixed wing aircraft. You can pitch up to climb, pitch down to descend. To transition to land, release shift to switch to hover mode. This slows the helo down to landing parameters. At that point, just flare off your airspeed an descend.
tl;dr – The helo flight mechanics are actually fairly grounded in reality. Watch for the nose to pitch up after takeoff, then hold shift and fly like a plane. Release shift to become more maneuverable and switch to landing mode.