Instead of going out and harvesting all the resources needed to complete the harvesting quests you can buy the resources instead or use resources you collected prior to receiving the quest. Here is how:
Start by getting the quest from the Harvesting trainer for the resource you want to harvest.
The quests have 2 stages, stage 1 they will ask you for 20 pieces of common resource. (ie. Copper, Gold, rough leather, etc...) . In stage 2 they will ask you for a single piece of a rare resource (ie. Blue Iron, Platinum, etc ..)
Stage 1
The quest requires that you gather and turn in 20 Items. (gold in this example) Go to the trader and buy 20 pieces of the "Tradeskill resource" Gold. check your post office and retrieve the gold from the mail attachment.
Now open your inventory and click the resource tab. This puts you into the resource inventory. If the gold is all in 1 stack, shift+ right click the stack and select split. Split 1 piece from the stack. Leave the 1 piece in your inventory and move the stack to your bank. Then drag the stack back onto the 1 piece in your inventory and let them re-stack. You should start receiving quest credit after the 1st or 2nd piece stack.
Take note of how many pieces you got credit for and any split off another stack of how many ever pieces you still need credit for. Again, move them to the bank and then move them back to your inventory and stack them on the existing resource there. Once you start receiving quest credit, you will continue receiving credit for any pieces added until the quest is complete.
Stage 2
This is basically the same as stage 1 but you only need credit for 1 piece. In order to do this you will need a minimum of 2 pieces of the resource you want. and some of them can be expensive to buy.
Start with the 2 pieces of the rare resource you need (ie Platinum). Put them in your resource inventory and split them as explained in Stage 1. Use the same system as in Stage 1, Split the stack and move to and from the bank and re-stack. After 2 or 3 times of repeating this you will get quest credit.
Remember that you also need to turn in the pieces you just got credit for to the trainer, so you can sell any extra pieces you are left with, but keep 20 of the common and 1 of the rare resources to turn in.