Here is the best no-risk, battle-free, get rich quick scheme in Skyrim.
Collect ingredients. Buy out the cheap stock of an alchemy merchant. It doesn't matter what you start with, as long as there's sufficient variety for the first "round."
At an alchemy lab, go to your full ingredient list and, beginning at the top, systematically "learn" their uses without wasting a single ingredient by using the following method:
Put your laptop next to you and find your first ingredient on this online ingredient's list and right-click on it. Open in new tab. This page shows all four potions that particular ingredient makes, along with each potion's full ingredient list.
Look for those ingredients with asterisks (*) on the web page, and if you have an ingredient marked as such, add it immediately in your game screen. You now have a potion with 2 effects, and have learned at least 3 uses for the ingredients. Add a third ingredient, ideally from another potion list, and now you've learned even more ingredient effects for one or more of each ingredient.
Go down your ingredient list from top to bottom until all 4 effects of each ingredient are learned. Only use ingredients until you see all four effects are learned, then leave the remainder of that ingredient in your inventory. Go down your list until you can't learn any new effects. Remember, the best potion to make is one in which you learn the maximum ammount of new effects per ingredient used.
Now, sell off your potions and move to the next alchemy merchant. Stay on the main roads and grab every ingredient you see along the way. Don't bother antagonizing large animals, just get to the next town, buy out the alchemy merchant's ingredient stock, and repeat the effect-learning process with everything in your inventory for which you still need to learn effects. One "round" down your ingredient list per town, then move on.
Don't worry about what potions you make, or how counter-intuitive the effects may seem. Even a potion with seemingly opposite effects will net you good money. The point for now is to systematically learn every effect and thus, every recipe. Using the web page to do this will be time consuming at first, but the learning curve rises exponentially due to the cross-referencing of effects. This rapidly leads to a state in which all potion recipes and ingredient effects are saved in your alchemy screen for future use, and you can quickly and easily create the most expensive (or useful) potions with no fuss or experimentation.
Circle Skyrim from town to town buying out each alchemy merchant's stock, creating potions in the store, and selling them back. After a few potion-making rounds, there will be no need to search the game world for specific ingredients; you'll be able to make almost anything with the standard stocks available in towns.
Along the way, your alchemy and speech will quickly increase to the point where you can obtain great alchemy perks and the all-important speech perk allowing you to sell anything to any merchant. Once you have this, you'll never lack willing buyers with money to burn.
Obviously, any enchanted item that increases alchemy or speech is a must-buy.
This is the best method I've found so far to A. make fast cash, B. level up the two most universally useful skills, and C. avoid insane levelling with multiple skills that cause your character to become an "all talk, no fight" character, incapable of defending himself against similarly levelled enemies. As others have aptly noted, this is the main problem with the "clear caves, mine ore, smith weapons, enchant the weapons, and sell them" money-making technique; you simply do to much and level up too many non-combat skills. And in terms of time expended, there is simply no comparison. Stay out of the mines.
Hint: since you will inevitably gain character levels due to your alchemy / speech increases, make sure you train the maximum 5 times per level in a combat or magic skill. It helps to have Faendel along for this purpose with his free archery training, but other combat trainers are available in the towns, and with all your money you'll have no problem paying them.
Of course, for sneaky, archery-sniper characters, there is no better way to start the game. By getting Faendel right off the bat and beginning this process immediately at the beginning of the game, it is possible to have a 50 archery skill (his max training level), speech and alchemy in the 30's or 40's, tens of thousands of gold in your purse, and a near-endless supply of powerful poisons for your arrows (paralyze, slow) and potions (invisibility, health regen) before you even set foot in a cave. Best of all, your character level will still only be in the 20's. Even playing on "Master" difficulty, you'll easily out-match your prey.
Last important tip: Perverse as this may sound, you'll want to avoid The Lover's Stone or anything else that enhances your levelling speed until you bank sufficient funds to really start the game right. How many round-the-board tours you complete before you actually set off for adventure is up to you, but remember, the important thing is to make some money and have a comfortable cushion. There's nothing more frustrating that finding something in a shop that would be perfect for your character at just the right time, and not having the jack to cover it.
Perhaps the best thing about this technique is its honesty; aside from relying on the web ingredient research, you aren't employing any cheats or glitches. Frankly, considering that alchemy is supposed to be an academic art, and you are putting in genuine time painstakingly going through each ingredient on your laptop, I find it a perfectly honorable way to go about learning the effects and recipes. No real alchemist would run around eating leaves and making random potions in order to learn their effects. He would research all available sources first, going about his work in a scientific manner.
As for Faendel's free training, my feeling is this: There are only 3 followers in the game that can train you, and the other 2 aside from Faendel are absurdly hard to obtain. The fact that followers give you free training is logical and was obviously an intended part of the game, not an exploitable glitch. Would a friend who fights at your side in real life charge you money to teach you a skill? I doubt it.
Some may disagree, but in my view, in RPG's (as in life) there is nothing better than having lots and lots of 'effin money. Sleep in the park if you want; I'd rather occupy an OFFICE on Wall Street.