In Farming Simulator 19, poplars can be planted on any field - you don't need to cultivate the field beforehand. To plant poplars, use the same planter that you've used to sow other trees: Damcon PL-75 (Tools->Forestry machines). You don't need to use a forklift to fill it up: buy a stillage with poplar seedlings and fill the planter as you would any spreader - approach the machine and press the filling button.
You sow poplars only once - cut down trees regenerate on their own; it should be mentioned that poplars' growth time is longer than most plants (2x).
When poplars are grown, it is time to cut them. You can utilize a forage harvester with poplar header, which will cut the upper parts of the trees and convert it into chaff. As the harvester doesn't have its own tank, you can attach a small trailer (e.g. HDK) or drive a tractor with a trailer next to the working harvester.
You can also utilize large sugarcane trailer (found in Sugarcane technology), but your employees may have problems with turning, leaving large areas of uncut poplars (nevertheless, your employees will be able to harvest the field on their own, almost without your input).
The resulting woodchips can be safely kept on the ground and loaded later using the belt system or a shovel; this is important, as the amount of woodchips per hectare may amount to even 80 000l (standard price on medium difficulty level is 180$; +30% on easy; -50% on hard).
To sell the woodchips, travel to a selling point (in this case, a lumber mill). Important: the woodchips are sold at the back of the lumber mill - there, you'll find a heap of woodchips where you need to tip your load. You'll get the money once the tipping process has been completed.