I mained Ashe for awhile she's one of my favorite characters and I play her often. I usually start with a
Doran's blade (it has a lot of bonuses, including health and damage, so it's a good starting item)
or boots and health and mana pots (mostly health pots so you can stay longer in lanes and boots for more mobility)
I really don't suggest getting AS before increased damage, because Ashe hits like a wimp early game. I tried getting AS first on Ashe after my first two items and I had fast attack speed but I could do a lot less damage than if I had gotten a BF Sword or another Doran's blade (if I'm farming slowly or I'm not getting gold fast enough, I get a second Doran's blade first).
What I usually try to get next is a BF Sword. It adds a lot of damage and it's a great item that gives you three other alternatives: bloodthirster , black cleaver(which adds AS /and/ damage, but get BF sword first before dagger), or infinity edge. I usually build up to a black cleaver first. Blood thirster has life steal, so I usually get that if there's a lot of physical damage from the other team.
But otherwise, I get infinity edge.
If I'm getting shot down too quickly, I get a banshee's veil for added health among a lot of other benefits. I might also consider a frozen mallet, but I never tried this in game.
If there are tanks, definitely get madred's bloodrazor and yumumu's ghost blade (which is great on Ashe).
For more survivability consider getting health and defense items that also give you added damage. It might not be as much as if you were having an Infinity Edge but at least you can survive.
I'm new too, so it's not definitely not perfect, but I find that it's effective for me because by end game, as a carry should, I'm usually doing heavy amounts of damage.
Oh, and here are some more tips as Ashe:
your passive allows you to do an early game combo where you hit the enemy with your first shot, and then volley them. That usually takes up to half their health or more.
in a team fight always, always stand in the back. i can't emphasize how squishy you are. i had to learn this the hard way plus by level 20 (i'm 23), people will be focusing you cause you're the main source of damage.
use your ult a lot, because it's got a very low cool down. it has a great benefit, it can stun, and therefore it can be used to get a kill or just to help in a team fight. it can shoot across the map but be careful about accuracy cause teammates usually rage when you miss by very little, and they end up dieing