A bucket of sap is obtained by using a knife with an evergreen tree, while having an empty bucket in the inventory. You cannot get sap from a maple tree. A knife on the tool belt will work; just use the empty bucket on the tree. One place to find evergreen trees is just north of the Falador lodestone, or right next to the Catherby lodestone.
One of the closest banks to evergreen trees is in Taverley, where the closest tree behind the bank is 16 squares away from the banker.
While many quests guides claim you must get sap from an evergreen, there are also many trees which give sap that are not described as 'Evergreen'. Usually they appear slightly taller and pointier than those surrounding them. A useful bunch of these trees may be found north of Falador or in Catherby near the fishing spots. The examine for these trees is: "A commonly found tree." instead of "One of the most common trees in RuneScape."
If you are having trouble getting sap from a tree, you might have to try around for a little bit, as not all Evergreens give you sap. This is evidenced by some of the Evergreens on White Wolf Mountain, however the trees next to the Bank of Taverley work fine.