There are a total of 257 collectibles (40 Artifacts, 71 Logs, 73 Weapon Parts, 61 Circuits, 12 Blueprints) in Dead Space 3. This video guide will show you all their locations. Blueprints are needed to develop new weapons. Weapon parts and Circuits can be used to upgrade your weapons.
Your collectible progress will be saved after reaching the next checkpoint. If you die or restart the last checkpoint you might have to collect some items again. To check out your progress you can go to chapter select and press triangle (PS3) / Y (X360). You are able to see the progress for each individual chapter. The game has 19 chapters in total.
Finding everything will earn you 6 trophies or achievements: Gun Collector, The Professor, The Librarian, The Armorer, There's Always Peng!, Aliens

IMPORTANT: To get all of the collectibles in chapter 4,11 and 14 you have to play co-op. Chapter 7 and 12 have no collectibles at all.