Arcadia is the expert level alchemy trainer in Skyrim and can teach you all the way up to level 75. The cool thing about spending money on her training is that she keeps that gold and it never resets. By this I mean if you spend 6000 gold on her training, the next time you restock her she'll have 6500 gold instead of the normal 500. This makes her training basically free since you earn all your money back from her by selling your potions. Just be sure not to get carried away with training an end up broke and back in Riverwood cutting wood for gold. Arcadia can train you a max of 5 times per level so be sure to have her do it to further speed up this step. Last, it's been reported that if you leave her shop the gold you gave her from training will go away. With this in mind, it's best to stay and restock her while in her shop while you're trying to recoup your training gold.
From here on out, it's up to you continue the processes above and get your alchemy skill all the way up to that elusive 100 level. If you want to make selling potions easier, you can invest four perks into speech until you get the investor perk. This perk will increase the gold limit of any vendor you invest in and the previous perk allows you to sell any item to any merchant. This means you can sell potions to the vendors outside of Arcadia's as well. This perk allotment though is by no means required, but it does give you a lot more places and gold to sell your potion stock.
Now that you've finished with your alchemy leveling it's time to start leveling Smithing and Enchanting. These two go up much faster than Alchemy so fear not, the most tedious part is behind you.
When I finished getting my alchemy level to 100 I had around 65,000 in gold from the process. This was more than enough to finish out the rest of these steps. The first thing you'll want to do now is go to Warmaiden's and talk with both Adrianne and Ulfberth. Buy as many iron ingots, iron ores and leather straps from them that you can. Next go to Belethors and buy all the iron ore, filled petty soul gems and lockpicks he has. Lastly head up to the Skyforge and talk with Eorlund Gray-Mane. Buy all the iron ingots, iron ores and leather straps he has as well. Once that's done head back to Warmaiden's and make iron ingots out of the ore you bought using the smelter. Now take a look in your inventory and make sure you have as many or more iron ingots as you do leather straps. If you don't buy leather and make the straps at the tanning hide.
At this point, head back to the Standing Stones and choose the Warrior sign. From here go to the Bannered Mare in Whiterun and purchase a room for 10 gold. Sleep for 12 hours and get the well rested effect. Now head back to Warmaiden's in Whiterun and begin smithing iron daggers. Iron daggers cost 1 iron ingot and 1 leather strap to make each. The problem you'll run into is that iron daggers weigh 2 units while the ingredients weigh just over 1. This means the smithing process will leave you over-encumbered. Don't fret though as Adrianne is always outside working so you can slither over to her once you've made as many iron dagger as you can. For reference I usually didn't start making daggers until I could make at least 100 in a sitting. Once you're done smithing the daggers, sell only as many to Adrianne as is required for you to be 10 units under your max carrying capacity.
As you begin smithing daggers your skill level will sky rocket quickly. At this point you have a choice of perks and it depends on your armor preference. The right side of the tree focuses on heavy armor while the left side focuses on light armor. Daedric armor, although it only requires 90 smithing, has a higher armor rating than heavy dragon plate. It's important to note that you can't go up the left hand part of the tree to dragon smithing and then choose daedric smithing. You must go up the right side to get that perk. Arcane smithing is a personal choice too. Since you're making your own equipment you don't really need it, but should you want to improve a magic item you find later on you'll need it. Last, remember the daedric hearts you were buying from Arcadia in the last step? Those will be required to create any piece of daedric armor or weaponry later on.
Now that you've got a ton of daggers it's time to begin enchanting. If you've followed my advice and bought the filled petty soul gems from Belethor you should have a good head start on this process. Before you can begin though you'll need an enchantment to actually put on the daggers. I recommend the banish, paralyze, turn undead or absorb health enchantments as those are more worth than other enchantments. Banish is worth the most but is hard to find though so of you see a paralyze or absorb health item just use that instead. To learn enchantment you must disenchant an item with the enchantment already on it. Warmaiden's and The Drunken Huntsman are the only places in town that sell enchanted weapons. This means you'll have to peruse the stocks of Adrianne, Ulfberth and Elrindir to look for the banish, paralyze and absorb health enchantments. If they don't have it, restock them and look again until they do. Once you have a bought an item with that enchantment, head back to the standing stones and choose the mage sign. From there fast travel to and enter Dragonsreach.
When you enter Dragonsreach head to the right of the Jarl (as you face him) and speak with Farengar. Buy all the filled lesser and petty soul gems he has. Now check your inventory to see if you have as many filled soul gems as you do daggers. If not, restock Farengar over and over and continue buying filled lesser and petty soul gems from him until you do. From here go to the Bannered Mare in Whiterun and purchase a room for 10 gold. Sleep for 12 hours and get the well rested effect. Now head back to Dragonsreach and go the the enchantment table by Farengar and disenchant the weapon with the banish, paralyze or absorb health enchantment. Once that's done begin enchanting all the iron daggers in your inventory. While doing this, begin investing in the following perks as soon as you can:
- Enchanter (All five ranks)
- Extra Effect (gained at level 100)
While you're doing this, you'll notice that the value of the dagger will begin to fall even though the enchantment is getting more powerful. I'm not sure why this happens or how to stop it but it has no effect on the experience you gain from each dagger.
Once you've enchanted all your daggers, it's time to sell them off. Adrianne, Ulfberth, Belethor, Eorlund Gray-Mane and Elrindir will all buy the daggers from you. While you're selling the daggers off begin buying all iron ingots, iron ore, filled petty soul gems and leather straps you can. If, after you've seen all the vendors, you still have enchanted daggers, it's time to restock the merchants and do the process above again. From here it's just a matter of doing all of the above over and over again until your enchanting and smithing skills are at 100. You'll hit 100 smithing before you hit 100 enchanting so you'll need to make a lot of daggers after you hit 100 just to have them for enchanting. Also, don't spend time getting filled lesser and petty soul gems from Belethor unless you're at his place to sell daggers and buy ore. Farengar stocks them much more regularly.