Is the particle up redundant in phrases like this one?
to pick up berries (flowers, fruit)
Or is it acceptable and doesn't change the meaning 'to gather' (update: I mean, to gather in the woods, in the garden)?
It is not redundant but unidiomatic.
We ordinarily speak of picking fruit or berries, without up, from the trees or bushes on which they grow; we use pick up only if they have been spilled (that is, we pick them up off the floor or other surface they have been spilled on) or if we are stop briefly at a shop or stand to obtain them.
If you're saying that you are going to actually collect flowers or berries, "up" is not only redundant, it's outright wrong.
We don't "pick up" things when we gather them, we "pick" them.
I spent the afternoon picking flowers in the field around my house.
I picked these berries this morning.
Now, you can "pick up" stuff... but it means to get them from somewhere and generally does not mean that you actually harvested them:
I have to pick up the flowers from the florist at 2 pm.
Remind me to pick up some blueberries when we're at the grocery store later, will you?
In this case, these items are already harvested, we're just saying that they need to be purchased or taken into possession.
It is neither redundant nor wrong, it just changes the meaning of your sentence.
"I am picking berries" means that I am picking berries directly from the plant.
"I am picking up berries" means that I am picking them up from the floor. Or the whole box of them from the supplier's truck.
In addition to the shades of meaning provided by others, when the meaning is to select or choose, it has to be "pick" rather than "pick up": "Please buy some apples, but make sure to pick the green ones".