A 1st level wizard starts by knowing all level 0 spells and a number of level 1 spells equal to 3 plus his Intelligence modifier, and he gets 2 more spells per level for free.
How does this work for a cleric?
Page 32 of the PHB has the description of the Cleric's Spells class feature.
So, yes, your cleric does not need to add his spells to a list of known spells, because (unlike the arcane casters in this book but like every divine class in this book) he already knows the whole list, expanded by the whole cleric spells list from any supplements that are OK in your game.
Don't trust the short descriptions on the list. Always delve deeper in the chapter, where all the spells are written down in extended form in alphabetical order, and make your daily choices based on the whole description of each spell.
To fully understand how spellcasting as a cleric works, I suggest you to also read the whole Spells paragraph and chapter 10, especially the Divine Spells section starting on page 179.
A side note: While there are some spells the Cleric can't cast because of alignment, he can still prepare those and they're known to him. Also, since he knows all the spells, some problems usually arise when knowing some spells is part of a prerequisite for a prestige class - usually, the DM rules that you only know spells of the levels you can already cast.