The best melee weapon for non-members is probably the Gravite Rapier, in terms of damage per second. The weapon with the highest accuracy and strength for nonmembers is the Gravite 2h sword, though this weapon attacks much slower. Please note that both of these weapons require you to dungeoneer for quite a while, especially if your combat level is 90 or higher.
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Full Rune is the current best melee Armour for free to play characters.
Well if you need a ranged weapon it is the bow and arrow, but the best melee weapon is the diamond sword, made with a stick with two diamonds on top.
In my opinion the best is a rune scimatar
vestas sword, but the best weapon is chaotic maul.
The overall best f2p Runescape weapon is actually a range weapon - Maple Longbow (sighted). It has a range bonus of over 100. The best f2p degradeable weapon is the Gravite 2h. Both are Dungeonneering weapons and need 45 Dungeonneering levels.