This can be done early in the game. More investigations unlock as you discover clues, and battle monsters.
Go to the RESOURCE CENTER next to the bazaar in Astera. Register several investigations to Slay 20 Vespoids (mosquito looking creatures). Each successful attempt is considered a completion toward 50. A ranged weapon is recommended, a bowgun or bow. At the nearby QUEST BOARD, choose an investigation to slay 20 Vespoids, and choose to spawn at CAMP 11 in ANCIENT FOREST.
Head to AREA 6 and you'll see plenty of vespoids. They continue spawning from the top of the trees up on the raised plateau. It takes a couple of minutes per run and is a fast option. It's a bit of a grind, but very obtainable.
BOUNTIES: Work on bounties as you go. There are mushrooms, bones, plants, bugs and honey in the area you can loot along the way and during the 60 sec wait after completion. Plus ARMOR SPHERES for every completion. Return to headquarters instead of camp to replace filled bounties.
SOS FLARES: Send up a flare at the start of each investigation. Others can join and quickly assist, and you can trade Guild cards (need 50), and get credit for a multiplayer Quest completion (need 100). This helps others get their 10 SOS assists as well. You'll be working on 5 achievements as you go.
OTHER OPTIONS: You can choose investigations that require you to defeat large monsters (which works towards slaying 500 monsters achievement), OR you can choose to respond to SOS flares from other players that are doing investigations, if you can find them (which works towards the achievement to assist and complete 10 SOS flares sent by other players).