There are two mines in the Lumbridge Swamp. The Eastern Mine is located south of Lumbridge, and there is a mining tutor who can give you some basic tips about mining. There are 5 copper ores and 5 tin ores here. This place is good for starters, and there is a furnace in Lumbridge to smelt the tin and copper ores into bronze bars.
For free players, the closest bank is the Lumbridge Castle. Members can use the agility shortcut to the Al Kharid bank with level 8 agility.