Collect nirnroots, just following shores of water. Nirnroots then can be exchanged in writs (mostly writs ask nirnroots), for other materials.
While you collect nirnroots you'd pick up a ton of water hyacint as well.

I havent noticed any difference between regions' herb spread out. Even in Alak'ir I found a lot of herbs. Though I found more dragonthorn there than nirnroot to be sure.

Lush area regions, forests I found quite good for picking larger ammounts of shrooms. Practically I cant mount my guar before I need to dismount again seeing a new pickable flower.
Keen eye in alchemy helps a lot.

Also.... if you want to build up a good amount of materials dont waste them for everyday use potions. Wait till you can unlock the alchemy passive that multiplies the potion output. Then you can brew en masse.

Side note: Namira rots, stinkhorns, whitecaps are pretty rare, search around ruins to get those.

For leveling alchemy, stock up on mushrooms of all kinds. You can usually buy them very cheap in guild stores. Reason to go for mushrooms is simple: You advance the same amount irrespective whether it's a "good" or "bad" potion you create. Mushrooms (except for blue entoloma and namira's rot) have mostly negative properties. The potions you create from them are more like poisons and not useful at all. Hence, these ingredients are dirt cheap, if yo go looking in stores you find them for as little as 10g per piece.

Since you will mass produce stuff to gain crafting levels, more than you can ever use while leveling your character, you would waste valuable herbs by using them. Research on the net a couple combinations using the negative property combinations of mushrooms, harvest about 100 to 150 of each solvent, craft until you can invest another point in the passive and upgrade to next solvent.

NB: Veteran Rank 1 potions have the same inspiration as Veteran Rank 5 potions, so there is no use in wasting those cloud mists for power leveling.

Especially Columbine, Mountain Flower, Bugloss, Blessed Thistle, Dragonthorn, Wormwood, Corn Flower, Lady's Smock, Namira's Rot and Blue Entoloma should bever be wasted to power level alchemy, since those are the main ingredients for the high end potions. As mentioned before, Nirnroot is often asked as part of alchemy writs, so keep those in stock as well.

You're welcome

I run the full perimeter of The Rift. By the time I make it back to Riften, I generally have at least 10-15 a stack for almost every plant, including ore and cloth raw mats. However, that's what I do and there may be better zones to do this in. This is just my go to for alchemy mat farming.