I am working on making a wizard fighter multiclass character where the wizard is the original class. How many spells should I have in my spellbook if I am a level 1 wizard and level 1 fighter? I have looked, but I could not find an answer and I am a little confused by what it says in the PHB.

As a Fighter1/Wizard1, you have 2 Spell Slots, and 6 Known Spells

There's two modes of determining your spell slots, based on how your multiclass levels are configured.

1. 
   If only one of your classes has Spellcasting capabilities, then you use the chart for that class as though you only have levels in that class. So for a Wizard9/Rogue7, use the Wizard chart at level 9. For a Sorcerer3/Monk7, use the Sorcerer chart at level 3.
   
   
   • Warlocks have a different kind of spellcasting feature called "Pact Magic", which means their spell slots are different from other spellcasters in multiple ways. As a result, Warlocks are not treated as Spellcasting classes, for the purpose of interpreting these rules; instead, you track spell slots for these two classes separately, and do not use the Multiclassing rules for Spellcasting.
   
   


2. 
   If more than one of your classes has Spellcasting capabilities, then you use the Multiclass Spellcasting chart found in the Player's Handbook.

Spell Slots. You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, half your levels (rounded down) in the paladin and ranger classes, and a third of your fighter or rogue levels (rounded down) if you have the Eldritch Knight or the Arcane Trickster feature. Use this total to determine your spell slots by consulting the Multiclass Spellcaster table.

[...]

beginarray
hline
Lvl.& 1st& 2nd& 3rd& 4th& 5th& 6th& 7th& 8th& 9th \ hline
1st&2&-&-&-&-&-&-&-&- \ hline
2nd&3&-&-&-&-&-&-&-&- \ hline
3rd&4&2&-&-&-&-&-&-&- \ hline
4th&4&3&-&-&-&-&-&-&- \ hline
...&...&...&...&...&...&...&...&...&... \ hline
20th&4&3&3&3&3&2&2&1&1 \ hline
endarray

—Multiclassing, Player's Handbook, pg. 164

All of these important rules, including the rules for Multiclassed Spellcasters, are found in that chapter, for your information.

In your case, because only one class has Spellcasting levels (the Wizard), you'd use rule 1, and simply use your Wizard level and the chart found under the Wizard class to determine your spell slots.

You Learn and Prepare spells for each of your classes individually, regardless of how many different spellcasting classes you have

This is also found in the Multiclassing rules:

Spells Known and Prepared. You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class. If you are a ranger 4/wizard 3, for example, you know three 1st-level ranger spells based on your levels in the ranger class. As 3rd-level wizard, you know three wizard cantrips, and your spellbook contains ten wizard spells, two of which (the two you gained when you reached 3rd level as a wizard) can be 2nd-level spells. If your intelligence is 16, you can prepare six wizard spells from your spellbook.

[...]

If a cantrip of yours increases in power at higher levels, the increase is based on your character level, not your level in a particular class.

—Multiclassing, Player's Handbook, pg. 164

So in your case, you'd continue to use your single Wizard level to determine how many spells you know, and how many you can prepare. If you had levels in other Spellcasting classes, you'd determine your spells for your Wizard individually, then for each of the other Spellcasting classes you have, and then your total known/prepared spells list would be the sum total of each class you have, with the Spellcasting modifier for each spell being based on which class you learned/prepared the spell from.

The number of Known Spells a Wizard has is campaign dependent

Just to make sure it's not left unstated: the number of "Known spells" for a Wizard is a little strange because their class features allow them to learn additional spells if they're given the time and money to copy down spells from a scroll or other source. So as a Wizard, you may very well know more spells than what the Player's Handbook says you should know at a given level. That's a feature that is particular to Wizards, and it is incumbent on you to make sure you understand that if you plan to play a Multiclassed character. Your DM should be able to assist in keeping your spellbook consistent.

6

The same as a 1st level wizard. You can add 2 spells anytime you gain a level in wizard. You can also add any you can copy from other wizard’s spellbooks or wizard spells on scrolls.

You need to understand the distinction between:

• spells in your spellbook,


• spells prepared, and


• spell slots.

At any given time you can prepare a number of spells from your spellbook equal to your Intelligence modifier + your wizard level (minimum of one spell). This will almost always be less than the number of spells in your spellbook. You can change any or all of these when you finish a long rest.

As a 1st level wizard you get 2 1st level spell slots so you can cast 2 of the spells you have prepared before a long rest or using Arcane Recovery. A long rest would get you back to 2. Arcane Recovery will get you back half your level rounded down (minimum of 1) back, so 1.

As a wizard, you can cast any spell that has the ritual tag even if you don't have it prepared so long as you have your spellbook - and they don't use a spell slot.

It’s all in the Spellcasting section under Wizards.