Max level 50 can be done several ways. Learning traits of plants - to learn the first trait, you can right click and use the plant item Practice combining plants with solvents - make pots using Creation button at the top right, at an Alchemy Station. Use poisons - poisons work well for practice as a solvent as they aren't as costly to use.

Skills and Ranks

Some passive alchemy skills have ranks. Solvent Proficiency (1-8) will upgrade the solvent you can use

Rank 0: Level 3 pots - uses Natural Water & Grease
Rank 1: Level 10 pots - uses Clear Water & Ichor
Rank 2: Level 20 pots - uses Pristine Water & Slime
Rank 3: Level 30 pots - uses Cleansed Water & Gall
Rank 4: Level 40 pots - uses Filtered Water & Terebinthine
Rank 5: Level 50 pots - uses Purified Water & Pitch-Bile
Rank 6: Level CP50 pots - uses Cloud Mist & Tarblack
Rank 7: Level CP100 pots - uses Star Dew & Night-oil
Rank 8: Level CP150 pots - uses Lorkhan's Tears & Alkahest

Keen Eye: Reagents (1-3) will allow you to view reagents to gather easier
Rank 1: See reagents from 20 meters away
Rank 2: See reagents from 30 meters away
Rank 3: See reagents from 40 meters away

Medicinal Use (1-3) helps potion effects last longer
Rank 1: Potion effects last 10% longer
Rank 2: Potion effects last 20% longer
Rank 3: Potion effects last 30% longer

Chemistry (1-3) provides extra potions per attempt which saves reagents and solvents
Rank 1: 1 extra potion per crafting attempt
Rank 2: 2 extra potions per crafting attempt
Rank 3: 3 extra potions per crafting attempt

Laboratory Use will allow you to use an extra reagent (3 instead of 2)

Snakeblood (1-3) reduces negative effects in potions
Rank 1: 50%
Rank 2: 80%
Rank 3: 100%

Alchemy Writs

Making a potion of health, stamina, or magicka, or a potion of ravage health/magicka/stamina. Will also require turning in 3 of the same alchemy ingredient.

3 Alkahest, 3 Lorkhan’s Tears, three of 2 different reagents (e.g. 3 lady’s smock and 3 corn flower); possibility of an alchemy master writ or an alchemist survey

Maximizing Master Writ drops:
maximize your skill to 50, have skill points in Solvent Proficiency at 8/8, learn all the traits for each flower, getting the achievement Botanist, and put a skill point into Laboratory Use. Tips: If you put skills points in Chemistry, you get more pots/poisons per crafting (saving you materials), To skill up, use up your excess poison solvents first. Need to be made by the character doing the quest. Bought, traded or reward pots don't count Regarding alchemy survey – if you don’t like the reagents that have appeared, you can get out of the field of view and come back and the reagents will have changed. Daily Alchemy Addon (https://esoui.com/downloads/info1899) will save you time making these.

more to come...