You should be able to pick everything assuming you’re playing current and not Classic. I assume you’re talking about this character so that would rule out Classic.
In current if you try to harvest a resource, like a herb, without sufficient skill you will usually loot several ‘fragments’ of that resource. You can combine 10 fragments to make one whole.
So if you have 10/300 and tried to pick a Golden Sansem for example, you would loot a few fragments instead of whole units. You don’t need to go to the trainer to learn specific skills to pick certain plants, you can just do it, and you get better results with higher skill levels.
Keep in mind that they’ve now segmented trade skills for each expansion. So if you go to Outland you will need to learn Outland Herbalism to gain skill from herbing in Outland. You can only learn a specific expansion’s skill levels in that expansion zone so you’ll need to find a Herbalism trainer in Outland to teach you Outland Herbalism.
The exception being Classic level and Cataclysm, as they share the same trainers.