So if I play with Ashe, is it still acceptable to worry about mana regen items? I spam my abilities when they're up, but it seems I can only do that if I have the regen items, otherwise I'm mostly just shooting stuff from afar but not really doing much, cuz I run out of mana really fast, even with a large pool.
Ranged Carries(Ashe, Caitlyn, etc) tend to rely on Auto-attacks for their damage output, thus building mana/regen is going to set back your damage. For the most part, it isn't really effective to spam your abilities(Ezreal tends to be a partial exception).
If you enjoy heavier use of abilities/combos, casters might be worth a look.
Also, some abilities have scaling. They'll show up as (+0) in a certain color:
Green = Ability Power
Orange = Attack Damage
Yellow = Armor
Red = Health
Dark Blue = Mana
White = Cooldown Reduction
Solomid.net is filled with streams, guides, and videos, too.