Q:

Multiclassing Cleric LvL 4 with Warlock lvl 1

I am having a nice argument with my buddy about multiclassing. I am new to the game so there is a lot of things I don't know, he is quite versed though. Our argument is centered in the amount of LvL 1, LvL 2 and LvL 3 Spells I can have. As my character is LvL 4 Cleric and LvL 1 Warlock, I assume he is LvL 5 in total. So checking the Multiclass Spellcasting Table, it says he has access to:
4 LvL 1 spells, 3 LvL 2 Spells and 2 LvL 3 Spells.
Counting that Warlock Pact Magic allows the character to use the Warlock spell slots as Warlock Spells or Cleric Spells, he says that the LvL I have from the Warlock doesn't allows me to have the amount of spells I quoted above, because I am still a LvL 4 Cleric, so I can't use LvL 3 Spells, neither use a LvL 2 Spell as a LvL 3. So the amount of spells I have according to the multiclass table is the one of a LvL 4:
4 LvL 1 spells and 3 LvL 2 Spells
Our main problem is related to how we mix Warlock to the Multiclass table.

A:

You don't use the multiclass spellcasting rules
The multiclassing rule for the spellcasting feature (which gives you spell slots based on the Multiclassing Spellcaster: Spell slot table) because it only comes into effect: 

Once
  you have the Spellcasting feature from more than one
  class, use the rules below. If you multiclass but have the
  Spellcasting feature from only one class, you follow the
  rules as described in that class.
--- Player's Handbook, p. 164

Notably, Warlocks don't get a Spellcasting feature, they get a differently named feature called Pact Magic. Thus your spell slots aren't replaced by the multiclassing rules, but instead added from your two sources of spell slots. This gives a Cleric 4/Warlock 1 character 5 × 1st level slots and 3 × 2nd level slots. 
Thankfully, due to the multiclassing rule on Pact Magic (also page 164 of the PHB) you can use the different slots interchangeably for casting, but note that the Warlock slot will recharge on short rests and the Cleric ones on a long rest. 

A:

If you look in the rules for multiclass spellcasting, it has a section dedicated to pact magic (which is what you have as a warlock)

Pact Magic. If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.

As a 4th level Cleric you have 4 1st level and 3 2nd level spell slots that replenish on a long rest. You can use these slots to cast spells from either class as long as you know them. And as a 1st level Warlock you have a single first level spell slot that replenishes on a short rest. You can use this slot to casts spells from either class just as you can with the slots you get from Cleric.

