Applications for tree sap are woven deep into history. Food, medicine, illumination, industry, etc. Extracting it shouldn't be complicated.
The first step is to craft a tap from a block of wood or a nugget of metal. Then the tap is driven into the tree, creating a tapped tree. Sap will accumulate in the tap as honey does in a bee hive. Metal taps will draw sap faster. The sap can be collected with a bucket periodically.
The quality and type of sap is dependent on the tree it was obtained from. Maple and birch sap can be boiled down in a caldron into syrup (1:40 reduction for maple and 1:>80 for birch irl, but I think 1:10 will suffice) and further into sugar. Conifer sap can be reduced into resin. Resin can then be distilled into rosin and turpentine, dissolved with linseed oil in vinegar or turpentine to produce varnish, or used as is for other applications such as torches.
Not quite so certain about other trees.