Fresh off Tutorial Island
You can stick where you are in Lumbridge (there are two oak trees west of the shop), or you can continue wandering around the realm, exploring and cutting as you go. If you choose to stay in Lumbridge, then continue selling the logs to the shop for the usual low price. However, if you decide to be a brave wanderer, then your best option is most likely to drop the logs as you cut them. Members also have the ability to cut Oak Logs outside of the Castle Wars mini game. With a bank nearby, it cannot be beat. Once you get to level 21, consider buying a Mithril Axe from a fellow player. At 1,000 to 2,000 coins, they may seem a little pricey.
You should start off with a Bronze Axe, given to you by the Survival Expert on Tutorial Island. Once you get into Lumbridge, and get some cash, you should find Bobs Axes to buy an Iron Axe, which is usually about 60 aion gold. Remember this place, for when you get to level 6, you will be able to buy a Steel Axe for about 200 coins. Now that you have your axe, its time to actually start cutting some logs. Since you are near Lumbridge, just walk north from Bobs Axes following the road until you find a general store. Just to the west of the shop there are plenty of regular trees to get you from level 1 to 15.
