Pickup will only activate if you win a battle, so no teleport.
Trace does not make the Pokemon pickup an item after battle.
It does work with berries used in battle, but if another berry is consumed in battle then you will no longer be able to pickup the other one.
If you have just gotten into Pokemon then you would want to know how it works in gen 8.
Its effects outside of battle are that the Pokemon with pickup has a 10% chance of picking up an item after you win a battle. What item you pickup depends on the level of the Pokemon. Generally the higher level the Pokemon the rarer items you can get.
There is a chart of all the items you can pickup here.
In battle it works slightly differently. This is quoted from bulbapedia:
At the end of the turn in which another Pokémon used a one-time use item (including throwing an item with Fling or using Natural Gift), a Pokémon with Pickup will collect the item if they are not holding anything. If more than one Pokémon has Pickup or Harvest, the faster Pokémon will restore an item first. If multiple items can be collected, the Pokémon will collect the item most recently consumed. Pickup cannot collect an item if the Pokémon that consumed the item is no longer on the field at the end of the turn, or if the item has already been recovered (such as with Recycle, another Pokémon's Pickup, or Harvest). Pickup cannot collect an item that a Pokémon consumed before switching out, even if it switched back in again before the end of the turn. Pickup cannot collect items that have been destroyed with Incinerate, eaten with Bug Bite or Pluck, or knocked off with Knock Off. A popped Air Balloon cannot be collected.
For more details in battle check this page.
Pickup is one of my favourite abilities, so I hope this helps.
