You can check your progress from the start menu. Diaries save upon collecting them so you can quit out to the main menu after grabbing a diary if you might have missed one.
The Metro Wiki has a nice written guide for all the diary's:
https://metrovideogame.fandom.com/wiki/Artyom%27s_Journal_(M...)
^If the link sends you to a blank Metro Wiki article try adding a closing bracket ")" manually. TA formats the link instead of leaving it as is.
Alternatively you can use this more in dept guide for more of a total walkthrough style:
https://gamefaqs.gamespot.com/xboxone/824052-metro-2033-redu...Disclaimer: When you start a new game it will reset your progress towards all the diaries so make sure to either grab them all in one go or go back into chapter select after finishing the game before starting your second playtrough if you don't want to have to collect all of them again.
If you prefer a video guide which I would highly recommend for this achievement then I would go with this one: