The only item's that allow a Pokémon to go first are the Quick Claw and the Custap Berry. It is a common misconception that the Quick Claw and Custap Berry will allow a Pokémon to go first once activated, that however is not the case as the Quick Claw and Custap Berry just enable a Pokémon to move first in it's priority bracket.
So for example if I give a Golem a Quick Claw and use Earthquake against a Lucario using Close Combat (the Lucario obviously being faster) and the Quick Claw activates, then Golem would go first.
However in the same situation if Golem uses Earthquake against Lucario using Bullet Punch and the Quick Claw activates again, then Lucario would go first because Bullet Punch is +1 Priority, while Earthquake is 0 Priority, thus both moves are not in the same Priority Bracket.
So to answer your question the Quick Claw and Custap Berry would enable you to go first if you use a move of the same Priority Bracket.
So for example if you sent out an Azumarill against a Talonflame with the Ability Gale Wings (which gives +1 Priority to Flying type moves) and you use Waterfall, while Talonflame uses Brave Bird, Talonflame would go first.
If you instead use Aqua Jet, while Talonflame uses Brave Bird, then since you're both using moves of equal Priority it's down to whoever's faster, which would generally be Talonflame, (unless you got one slow Talonflame and one extremely fast scarfed Azumarill).
If you give Azumarill a Quick Claw to hold and use Aqua Jet, while Talonflame uses Brave Bird and the Quick Claw activates, then you would go first since both Aqua Jet and Brave Bird are in the same Priority Bracket (+1) thanks to Gale Wings.
However using an item on a Pokémon will always go before a move is used, as using an item is +6 Priority and there's no move that has higher Priority than +6 (not counting Pursuit because that's only +7 when used against a target that switches out on the same turn).
So for example if I was battling Lance and I used an attacking move and he used a Full Restore, the Full Restore would restore his Pokémon's health before I could do any damage to it.
Using items and switching Pokémon will almost always precede a Pokémon's move.
