Sunshine daydreams guide to efficient alchemy leveling (google) will give you the order to discover traits that uses the least ingredients.
From there:
Don’t use more than two reagents. Three doesn’t make the xp better.
Use the best solvent possible, potion or poison is the same. Spend skill points in the first passive. If you want to get them back later, that’s fine, but it saves a fortune to level up alchemy as you grind it.
You can always look in guild stores to see what’s cheapest for grinding. I always use spider eggs and the things that pair with them because they are pretty easy to get.