Do I need one focus per spellcasting class I have, or does one work for both? In other words if I multi class from Warlock to Sorcerer, do I need an Arcane Focus for both classes or use just one between them?

An arcane focus is...

...a special item designed to channel the power of arcane spells. A sorcerer, warlock, or wizard can use such an item as a spellcasting focus, as described in the Spellcasting section.

There's an argument to be made, based on a passage in the multiclassing section, that might go counter to this:

Similarly, a spellcasting focus, such as a holy symbol, can be used only for the spells from the class associated with that focus.

However, read as natural language, any arcane focus is "associated" with all the arcane casting classes listed in the description of the arcane focus itself. The description of an arcane focus is where that association is made.

There is nothing to indicate that one arcane focus is functionally different from any other, nor to indicate it varies by class. The equipment lists do not include "wizard focus", "warlock focus", or "sorcerer focus", just...

...an arcane focus.

The PHB is ambiguously written and requires DM interpretation

There are three relevant sections to make this determination.

The Sections on Spellcasting Focuses and what those focuses are suggest that this is fine

Firstly, the sections for each class that specify what a class may use to cast spells. I'll use the Sorcerer class as an example, but all three classes have simple, identical wording for their "Spellcasting Focus" section:

Spellcasting Focus

You can use an arcane focus (found in chapter 5) as a spellcasting focus for your sorcerer spells.

Substitute "Warlock" or "Wizard" for "Sorcerer", and you have the wording for those other classes.

The second important section is referenced by that paragraph. In Chapter 5, "Equipment", under the "Adventuring Gear" section, the entry for Arcane Focuses reads like this:

Arcane Focus. An arcane focus is a special item—an orb, a crystal, a rod, a specially constructed staff, a wand-like length of wood, or some similar item—designed to channel the power of arcane spells. A sorcerer, warlock, or wizard can use such an item as a spellcasting focus, as described in chapter 10.

And this paragraph references the "Material Components" part of the "Components" paragraph in the Spellcasting chapter.

By these descriptions alone, one could easily be led to believe that an Arcane Focus should be transferable between these three classes, no caveats or exemptions to worry about.

However, the Multiclassing rules muddy this interpretation

The final important section is in the Multiclassing rules. In Chapter 9, where the Multiclassing rules are written, the paragraph on Spell Preparation has this to say (emphasis mine):

Spells Known and Prepared. You determine what spells you know and can prepare for each class individually...

[...]

Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell. Similarly, a spellcasting focus, such as a holy symbol, can be used only for the spells from the class associated with that focus.

The use of the singular word 'Class' seems to imply that one focus may only be used for one class, and without a clarifying sentence or paragraph implying that a focus compatible with multiple classes would be valid for all of them, we can't infer that it's therefore okay for an Arcane Focus to be applied to all of these classes.

One observation we can make is with its example of a Holy Symbol: in the Player's Handbook, a Holy Symbol is a valid focus for both Clerics and Paladins. Since this kind of focus could easily be used for two (relatively complementary) classes, the omission of an explicit clarification (i.e. a rewording that reads like "a spellcasting focus, such as a holy symbol, may only be used for the spells from a class that is permitted to use that kind of focus") and the decision to use a singular word rather than a plural, makes it impossible to affirmatively assert that a multiclassed Cleric/Paladin could use the same Holy Symbol for all their spells.

As far as I'm aware, Jeremy Crawford has never made a ruling on this topic (please leave a comment on this answer if he has!), and we don't have errata clarifying this issue so we're left with several conclusions:

• Until we reach the Multiclassing rules themselves, there's nothing that infers that a focus could not be shared between classes that are permitted to use the same object


• The Multiclassing rules themselves are Optional Rules, and thus subject to some degree of flexibility,


• 
  But, the Multiclassing rules themselves are not written to strictly support the allowance of one focus for multiple classes


• And, the Multiclassing rules are generally used to supercede rule interpretations that would otherwise be made in their absence—in other words, the Multiclassing rules are the reason why certain features that read "at level X gain the ability to..." are generally interpreted to actually mean "when you reach level X in this class gain the ability to...".

So as it is, you'll need to ask your DM how they would rule.

As usual, Component Pouches are the "universal focus", so to speak

Since every spellcasting class can use a Component Pouch to access their Material requirements, this is always an option to ensure that a multiclassed character can access all their spells without juggling a focus, and this remains true regardless of how your DM rules on this issue.

Yes, as specified above by the others.

Also it's worth noting that if one focus is specified on two different lists, then it works for both. A Yew Wand works for both Druids (Yew Wand is specified for Druid focus) and for Wizards (Wands in general work for wizards) Therefore, that same wand could be used for both types of spells by a dual-classed wizard/druid.