What does Time Machine back up?

Does Time Machine backup system files, kernel files and other OS-related files?

I have corrupted system drivers and files and I want to do a fresh install of OS X but I also want to back up my applications, files and folders.

I wonder if I should go with Time Machine or manually backing up each file.

I think I can comfort you with the following statement on the website of Apple (Update: The information cited here has been removed, but you can still find it in this archived version)

Time Machine automatically backs up your entire Mac, including system files, applications, accounts, preferences, email messages, music, photos, movies, and documents. But what makes Time Machine different from other backup applications is that it not only keeps a spare copy of every file, it remembers how your system looked on any given day—so you can revisit your Mac as it appeared in the past.

On the given page, there are even tutorials on how to restore your Mac from a Time Machine copy.

Please be aware those backups are not bootable. So you still have to manually install a fresh OS (from USB, DVD or other...)

Time Machine backs up EVERYTHING! That is why I recommend to use Time Machine to back up your system. If you decide to do a fresh install and only want to restore specific files or folders (like some apps, the documents folder, some music, but not the downloads folder...) you can do this via Time Machine too. You are not limited to take it all or nothing! You can pick what you like to restore, assuming you know where it is on your HDD.