Peas-N-Carrots Pistol - Unlocked from the start.
]Portable Pizza Oven- Unlocked from the start.
Babel Gun - Requires you to kill 24 tanks.
Blatherskite Crossbow - requires you to finish level 7 on Endless Mode. Use TooYoungToDie perk, and a weapon you feel comfortable with (Egon's Pride is a good choice), and you'll have no trouble. Also, at the end of level 5 pick the first elevator and you'll start from level one again. This makes it easier, because The Maw is considerably harder.
Egon's Pride - requires you to complete a level under par time. This is easier than it sounds: start a game, and just run for the doors and shoot them to open until you reach the first boss. You don't need to kill more enemies than the boss to complete a level. The gun unlocks when you reach the elevator at the end, if you are fast enough. You can use a Caketown run for this, because the starting items will make you stronger.
609mm Handcannon - is awarded to you by completing a level without taking damage. Very easy. In your runs number 5, 10, 20, 30, 40, 50 and 100, you'll start in a small level without enemies named CakeTown. Just reach the lift at the end of it without shooting yourself or falling off the edge and you're set.
Kegerator - requires finding 5 secrets in one stage. Lots of secrets require you to do a fair amount of jumps, so starting with the Bluegrass perk is a good idea. Also you can do an endless mode run, and after a few levels you'll have improved jumo height and such, which help a lot. Secret areas tend to be on the highest part of the rooms, or behind some walls that can be trespassed.
http://steamcommunity.com/sharedfiles/filedetails/?id=248552...Mini LHC Gun - requires you to discover 22 different items. This refers to pickup upgrades, gun mods, secondary weapons and usable items. There's no way to speed up this, as loot is random, so keep doing runs until it unlocks. CakeTown, and a random secret area called Hugbot Alley that you can access sometimes if you don't kill Hugbots can help a little.
Consolation charger - requires you to die 3 times. This will probably be your first unlocked weapon.
The Hedgehog - requires you to kill 250 spikeball launchers, that are small turrets with a few spikes on their base. They shoot spiked balls that bounce around, so you'll recognise them quickly. The only way to unlock this is doing several runs killing every spikeball launcher that you see. They usually spawn in groups of 3 up to 9..