General Achievements are those that do not fall under any other category

Achievements for getting gold
Just by picking up gold, you will get this achievement. Loot everything you see, open all the chests. Even if the gold amounts seem small, they add up. Treasure Pygmies/Treasure Bandits are a good source of gold. Each of these achievements is worth 10 points.

Scavenger: Pick up 10,000 gold (You might get this one as early as act I)
Spare Change: Pick up 50,000 gold. (Achievement Reward: Banner Shape)
Moneybags: Pick up 100,000 gold (Achievement Reward: Banner Accent)
Deep Pockets: Pick up 500,000 gold (Achievement Rewards: Banner Sigil)
Staying Gold: 1,000,000 gold (Banner Pattern)
All That Glitters: 5,000,000 gold (Banner Accent)
Spoils of War: 10,000,000 gold
The One Percent: 100,000,000 gold (Banner Accent)

Some of the above achievements will require many playthroughs for you to get them. Remember that achievements are shared per account, so you can work towards them with all your characters, and when the combined gold pick up reaches the achievement prerequisite, you will earn that achievement.

Breaking Not so Bad (10 Points)
Use a merchant to repair an item

You will get this one pretty early in the game. Your armor loses durability over battle, and when you die. Just go to any merchant and repair your items under the repair tab, to get this achievement.

The Comfort of Strangers (10 Pts)
Achievement Rewards: Banner Pattern
Recruit the Templar, Scoundrel, and Enchantress
Note that you have to actually have them join your team for you to earn this achievement. If you chose not to on the first opportunity, they will be waiting for you in town so you can just go and hire them.
The Templar is found in Act I at the Tristram Cathedral Level 3
The Scoundrel is found in Act I in the Fields of Misery, before the Drowned Temple
The Enchantress is found at the beginning of Act II in the Sundered Canyon

Armed to the Teeth (10)
Achievement Rewards: Banner Accent
Equip all inventory slots with items that have a minimum required level of 25 or higher

Prepared for Battle (10)
Equip all inventory slots with items that have a minimum required level of 60 or higher