By fighting. There is really no other way to gain significant amount of experience in any fighting-related skill.
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Well, you just kill monsters. Put your attack into defend mode, and then you will get defence exp. Check what is your denfence level in the skills page.
I attacked moss giants in the Varrock sewers until I got to level 60 attack. They have good drops, especially for members. Also, attack red spiders in the Varrock sewers or Hill Giants in the building next to Barbarian Village. Completing quests also give you experience in some areas.
kill 1000 lvl 5 goblines
463,885exp
All dragon weapons require LV 60 Attack to weild. However some might require some quests to be completed to weild. At any level, you can purchase Dragon Weapons.