The caveeat to that 10 minutes claim is, that you cant do it in the beginning, because you wont be properly prepared. And having the correct ingredients is the key. So after you got enchanting to 100, you very likely have banish, which means, you can make a very quick and easy profit on your enchants without leaving whiterun. (just make iron daggers with mats bought from the warmaiden, then buy filled soulgems up to common from the jarls mage, enchant with banish and paralyse/absorb health, sell, repeat)
Now, for that list of alchemy ingredients. The most important thing to look for everywhere is a giants toe, since it seems to put a serious spike into potion values. The more you have, the better. Whenever they refresh, plow through the giants camps, buy it from vendors, or steal it in the archmages quarters in the winterhod college, which is a great place for ingredients btw.
The reason for this is, in case someone didnt know, for alchemy leveling the value of the potion you create is the most important aspect. At alchemy level 50 a pot with Creep Cluster, Giants Toe and Wheat gives you a full skill level, and I still havent seen a more valuable potion.
There is a website, which lists the combinations of ingredients and their values. The value itself on this website is irrelevant, its about the proportion. Pick the most valuable from these lists (best value there is from 550-720) and buy the ingredients.
As a reference. At alchemly level 50 with the affordable perks and alchemy gear the potion I made was worth about 3000 (a third for selling)
http://www.endoftheweb.se/skyrim/index.php
But you dont have to go there, because I already did the work for you to break down a list of most valuable potions and the ingredients you need to look out for:
Value
(Giants Toe Creep Cluster Wheat) <- highest value potion
Hanging Moss Glow Mushroom Glow Dust 726
River Betty Large Antlers Giants Toe 704
River Betty Large Antlers Creep Cluster 704
Spriggans Sap Glow Mushroom Glow Dust 667
Swamp Fungal Pearl Briar Heart 657
Vampire Dust Luna Hawk Feathers 644
Vampire Dust Chaurus Egg Garlic 640
Imp Stool Briar Heart Slaughterfish Scales 634
Scaly Phot. Mora Tap. Creep Cluster 633
Spider Egg Bear Claws Canis Root 620
Hanging Moss Bear Claws Saber Eye 620
Spider Egg Canis Root Bear Claws 620
Spider Egg Canis Root Hanging Moss 620
Vampire Dust Luna Skeevers Tail 620
Nordic Barnacle Vampire Dust Chaurus Egg 619
Nordic Barnacle Vampire Dust Luna Moth 619
Namiras Rot Vampire Dust Luna Moth 619
Vampire Dust Chaurus Egg Nordic Barnacle 619
Vampire Dust Chaurus Egg Namiras Rot 619
Vampire Dust Luna Chaurus Egg 619
Vampire Dust Luna Hanging Moss 619
Vampire Dust Luna Butterfly Wing 619
Swamp Fungal Tundra Cotton Briar Heart 612
Hanging Moss Bear Claws Nordic Barnacle 591
Hanging Moss Bear Claws Namiras Rot 591
Hanging Moss Bear Claws Luna Moth 591
Hanging Moss Bear Claws Hargraven 591
Hanging Moss Bear Claws Chaurus Egg 591
Salt Pile River Betty Scaly Phot. 578
Salt Pile River Betty Creep Cluster 578
Spider Egg Glow Dust Glow Mushroom 573
Spriggans Sap Nightshade Glow Mushroom 571
Hanging Moss Glow Mushroom Nightshade 571
Salt Pile Garlic Large Antlers 567
Purple Mountain Bear Claws Hanging Moss 561
River Betty Large Antlers Whisp 551
River Betty Large Antlers Hawk Feathers 551
Scaly Phot. Mora Tap. Whisp 548
Scaly Phot. Mora Tap. River Betty 548
Scaly Phot. Mora Tap. Hawk Feathers 548
Scaly Phot. Mora Tap. Giants Toe 548
Salt Pile Jazbay River Betty 543
Salt Pile Jazbay Large Antlers 543
Salt Pile Jazbay Deathbell 543
Salt Pile River Betty Taproot 543
Salt Pile River Betty Dwarven Oil 543
Salt Pile Deathbell Taproot 543
Salt Pile Deathbell Jazbay 543
Salt Pile Deathbell Dwarven Oil 543
Dragons Tongue Fly Amantia Creep Cluster 540
Giants Toe Creep Cluster Wheat 514
Vampire Dust Luna Moth Wing 560 (added this, as it has a great value for just 2 ingredients, but when you work down the list, you most likely wont have vampire dust left anyway)
So the list of ingredients to ALWAYS buy and pickup is:
Bear Claws
Briar Heart
Butterfly Wing
Canis Root
Chaurus Egg
Creep Cluster
Deathbell
Dragons Tongue
Dwarven Oil
Fly Amantia
Garlic
Giants Toe
Glow Dust
Glow Mushroom
Hanging Moss
Hargraven
Hawk Feathers
Imp Stool
Jazbay
Large Antlers
Luna Moth Wing
Mora Tap.
Namiras Rot
Nightshade
Nordic Barnacle
Pearl
Purple Mountain
River Betty
Saber Eye
Salt Pile
Scaly Phot.
Skeevers Tail
Slaughterfish Scales
Spider Egg
Spriggans Sap
Swamp Fungal
Taproot
Tundra Cotton
Vampire Dust
Wheat
Sorry, if the formatting is a bit off.
Now, the only unexplainable things is, you find the potion with Giants Toe, Creep Cluster and Wheat, which I was talking about at the end of the list on this website, but there might be a bug, because this is the most valuable. You start out making this one and work your way down the list.
Now the preparation.
1. Choose the thief guardian stone for 20% faster leveling in alchemy. Go to your home and sleep for a 10% rested bonus (15 if married), or in an inn, if you dont own a house. This will speed up the process noticably.
2. Make alchemy gear. If you have 100 enchanting, you can drink a store bought potion to increase enchanting 25%. With that you can make 4 items to increase you alchemy skill 27% each for a total of 108%.
3. perks
I would expect, if you want to max your alchemy, you probably want to use the neccessary perks, which are 5x Alchemist, 1x Physician, 1x Benefactor. Of course you wont be able to pick up all of them as you are just starting to level alchemy. BUT make sure that whenever you reach the needed skill level, you stop making potions, pick up that perk and then go back to make potions, because they will be worth more and you will advance faster.
4. If you have the money, use it to train the 5 skill levels at each global level in arcadias. You'll get your money back in no time. When I was done with the 10 minutes I had pots worth more then 150k store selling value.
Thats pretty much it. Make sure you have the leveling bonuses, wear the gear, buy 5 skill levels every time you can, and make a pause everytime you can put another perk into alchemy.
I was level 50 alch when I started this just by playing around with some ingredients at some point earlier in the game, but if you are much lower, the levels will jump even faster.
At level 50, the giants toe potion gave me over a level each, but sadly I had only accumulated 15 giants toes, so at about level 65 I worked my way down the list with the values decreasing.
To give you an idea, in total I probably had to make less then 100 potions to get to 100, and since you often make 10-15 potions with the same ingredients you can imagine that it was rather quick.
So, sorry, its not really 10 minutes to get you to 100 if you just started the game, but on the other hand I wouldnt advise you to do that anyway. Start doing this when you are around 30, it will take you up 5 or 6 total levels and by that time you will have the needed ingredients to make this easy going.
Hope it helps.