Complete the game to unlock the New Game+  Mode. In this mode your experience level, weapons and upgrades carry over.  Additionally, you can build your structures in any order in the New Game Plus.

Though it says the requirement for a gold medal is 2 or less hits, you can actually be hit 3 times and still receive the gold medal.

If you upgrade the bow to have +50% damage (the second bow upgrade), you can destroy the targets in one shot, without the need for a power shot. This makes the gold medal much easier. If short on upgrade materials, a bronze medal will give you one something stringy upgrade material.

When having trouble with the Slinger Range Proving Ground, keep in mind that you can't get negative points. So, shooting a little bit before the first target shows up can get you a couple extra points that can make all the difference when going for a gold medal.

The Repeater (Crossbow) you get near the start of the game is awesome once upgraded with homing shots that are very helpful. Stick with it!

Throughout The Kid's adventure in his post-Calamity world, you'll come upon special items that he can used in the Forge to upgrade an assortment of weapons. The following materials are used for the corresponding weapons:

MATERIAL - WEAPON
Something Heavy - Cael Hammer
Something Stringy - Breaker's Bow
Something Nasty - Fang Repeater
Something Sharp - War Machete
Something Coarse - Scrap Musket
Something Greasy - Dueling Pistols
Something Pointy - Brusher's Pike
Something Fancy - Army Carbine
Something Burnt - Galleon Mortar
Something Foul - Fire Bellows
Something Wrong - Calamity Cannon

ACHIEVEMENT - HOW TO UNLOCK
The End (20 points) - Complete the story.
The Stranger (10 points) - Complete the Wharf District.
Pet Sitter (10 points) - Get four different domesticated creatures in the Bastion.
Vigilante (10 points) - Complete at least 50 percent of the Vigils in the Memorial.
Altruist (20 points) - Complete 100 percent of the Vigils in the Memorial.
Calamity Kid (30 points) - Complete the story in New Game Plus.
Mind Voyager (20 points) - Complete each trip to Who Knows Where.
Kid-at-Arms (10 points) - Use the Forge to apply at least one upgrade to every weapon.
Additionally there are four secret achievements.
The Ura (20 points) - Stop the siege on the Bastion.
The Beginning (20 points) - Complete the story, embracing the New World.
The Survivor (10 points) - Complete the Hanging Gardens and speak with the Survivor in the Bastion.
The Singer (20 points) - Complete Prosper Bluff and speak with the Singer in the Bastion.