Alchemy Specializations in War Within can be confusing at first. This guide will give you a solid, all-around build that works well for most playstyles, so you don't waste your Knowledge Points.
I'll guide you through the most efficient way to spend your points and highlight a few alternatives if you want to adjust the build to fit your playstyle.
Alchemy Specializations
Alchemy has four specializations. You unlock the first at 25, the second at 50, the third at 60, and the last one at 75.
1. Alchemy Master
This is the general Alchemy tree. The central node provides a large Skill bonus to all Alchemy crafts, and there are five sub-specializations, each tied to one of the new War Within herbs.
2. Fantastic Flasks
Pretty straightforward. This specialization increases your Skill and Resourcefulness when crafting Flasks and Phials.
3. Potent Potions
Same as Fantastic Flasks but for Potions. Increases your Skill, Multicraft, and Resourcefulness when crafting Potions.
4. Thaumaturgy
Thaumaturgy is the transmutation specialization, allowing you to convert basic materials into different ones. However, transmutations have low profit margins in War Within, making this spec much weaker for passive gold-making compared to Dragonflight.
Recommended Specialization Build
Since you'll unlock all four specializations, the real challenge is spending your Knowledge Points wisely. This build is a strong choice for most playstyles, whether you're crafting for yourself, taking advantage of the extra Flask duration for Mythic+ and raids, or selling consumables later on.
First 15 points for Mythic+ and Raids (Optional)
If you do a lot of raiding and Mythic+, I recommend spending your first 15 points in 
Fantastic Flasks. This increases the duration of your Flasks and Phials by 100%, meaning you'll only need to refresh them every 2 hours instead of 1. Over time, this will save you a lot of gold on consumables.
Fantastic Flasks
Alchemy Master (50)
Spending 50 points in Alchemy Master gives you +60 Skill for every Alchemy recipe. This makes it much easier to complete Patron crafting orders using lower-quality materials and without needing Concentration. It also reduces the amount of Concentration required to craft Quality 3 versions of any recipe.
Alchemy Master
Fantastic Flasks (30), Potent Potions (30)
I'd recommend finishing Fantastic Flasks and start putting points into the Potent Potions specialization.
Fantastic Flasks, Potent Potions
Getting a Guaranteed Quality 3 Potion or Flask
At this point, you probably have most Alchemy recipes from your daily experimentation. I'd start focusing on one Flask or Potion that you craft the most and want a guaranteed Quality 3 without using concentration.
So, using 
Tempered Potion as an example. It uses 
Mycobloom, 
Luredrop and 
Arathor's Spear as reagents. You'd need to put 40 Points into each sub-specialization inside the Alchemy Mastery Tree, so you would end up with something like the build in the picture below.
If you pick a Potion and not a Flask, you should definitely put points into Mycobloom Lore first and then the rest of the herbs. Mycobloom is used in every single potion, so you will get extra skill points for every potion, and guaranteed Quality 3 Healing Potions.
Spending the Rest of your Points
You can either focus on Alchemy Mastery to guarantee Quality 3 in all potions and flasks, or invest in Fantastic Flasks and Potent Potions for Resourcefulness and Multicraft. By the time you reach this stage, you'll likely have a good idea of where you want to spend the rest of your points.
I'd personally go for the 
Bulk Production tree in Potent Potions first and then Bulk Production in Fantastic Flasks, depending on which recipe did you pick in the previous step.