Just a heads up: farming souls is going to make your soul memory increase, making summons and invasions much more difficult. Don't fret, though, as soul memory does not affect the online component in NG+ or above, just your soul level.

1st Farming Area: Undead Crypt

If you've been in Undead Crypt, you probably know about those annoying, respawing ghosts. And if you've played Dark Souls 2 at all, you're bound to know that enemies in the game despawn after they're killed a certain amount of time. The thing is, the ghosts will respawn infinitely. All that is required of you, is to ring one of the bells. An undead will usually do it for you, but you can also do it yourself. Ringing a bell will make the ghosts spawn from their gravestones, to which you can then murder them and repeat the process. On new game, I was getting about a 1,000 souls for each kill. Just be careful not to destroy the gravestones, as that will prevent the ghosts from spawing. Don't worry, though, as there is a bell you can ring on the path to the area boss, which will spawn ghosts from towers that can't be broken.

The benefit to this farming technique is that you won't need to use any bonfire ascetics, allowing you to keep the game at the base level of difficulty (if you so desire). Here's a link to the video that explains it better than I do.

2nd Farming Area: Memories of the Ancients

Oh boy. This farming area is so good that it might be patched at some point. Essentially what you want to do is burn an ascetic on the 4th bonfire in the Forest of Fallen Giants. This will bring all the Giant memories into NG+ (and so on). Not only will this allow you to nab the following items: 2 Twinkling Titanite, 2 Bonfire Ascetics, 1 Soul Vessel, 1 Divine Blessing, 2 Fire Seeds, 5 Smooth $ Silky Stones, 1 Petrified Something, 3 Souls of a Great Hero, 3 Wilted Dusk Herbs, 3 Souls of a Hero - it will also allow you to fight the Giant Lord boss again. Which if you tackle with a ranged character (which you can simply change to with a soul vessel, as you will have plenty left over), becomes very easy, as you can stand on the platform to the left and easily dodge his one ranged attack. Killing the boss will grant you a +2 Ring of Giants, a Giant Lord Soul, and 200-300k souls depending on how many bonfire ascetics you have burned.

If you want a more detailed explanation, a reddit user created the guide.

I haven't found a need to farm in this game.

Honestly I'm doing it for the Twinkling Titanite at this point. I've got a few armour sets that I want to max out. And those smooth & silky stones can earn you some petrified dragon bones, which are hard to come by. Otherwise I'm sitting at over a million souls with nothing to spend it on.

Ya farming for twinkling and dragon, maybe a few other items. I guess I was just thinking purely of souls.

I farmed out the Shrine of Ashara last night for a butt load of twinkling titanite and human effigies.

Yeah, Shrine of Ashara is where I farmed out twinkling titanite. And since twinkling titanite was so scarce in the early portions of the game I used mostly gear that requires standard titanite. I've got 11 TT in my inventory and no idea what to use it on.

@mosespippy: Only 49 more and you can fully upgrade the best armor in the game!

@fredchuckdave: Best in what sense? Most total defense, most physical defense, most elemental defenses, or highest defense to weight ratio? How about something close to that but with a bonus effect? It's difficult to classify one armour as best.

@mosespippy: That aside, it takes 60 TT to upgrade a full set of armor. 15 per piece.

I would farm the dragon aerie for dragon & twinkling, too. Souls are extremely plentiful just by playing the game, often leaving me with souls I don't even know what to do with. However upgrading materials are certainly a rarer commodity, and the dragon aerie proves very useful to gaining a lot of dragon stones in particular due to the innumerable crystal lizards.