The rules for the Multilevel Spell Checker in the PHB (p. 175) or the Basic Rules clearly state how known and prepared spells work when subdivided into two or more classes (or subclasses) using the Spell Check feature:
Your ability to spellcast depends in part on your combined language levels in all your spellcasting classes, and partly on your individual language levels in those classes. If you have the magic function of more than one class, apply the following rules. If you have multiple classes but only use the magic function of a class, follow the rules described in this class.
Spells known and prepared. You decide which spells you know, and you can prepare for each class individually as if you were a one-time member of that class. For example, if you're a Ranger 4 / Wizard 3, you know three first-rank Ranger sayings based on your Ranger-level tiers. As a third-level wizard, you know three Wizard Cantrips, and your spellbook contains ten spells, of which two (the two you received when you reached the third level as a sorcerer) may be second-level spells. If your Intellect is 16, you can prepare six spells from your spellbook.
Each spell you know and prepare is linked to one of your classes, and you use the spell casting abilities of that class when casting the spell. Similarly, a magic effect focus, such as a sacred symbol, can only be used for the spells of the class associated with that focus.
As your Cantrip increases in strength at higher levels, the increase is based on your character level rather than your level in a particular class.
As explained above, if you create multiple spell classes, you will determine your known and prepared spells as if they were classified as a single class in each of these classes.
Note that some class / subclass features (such as the extra magical secrets of the Overlord Barber, Cleric Domain Charms, or Paladin Oath Spells) may contain additional known or prepared spells that are not counted against your number of known / prepared spells for that class , In this case, you can use the class / subclass function description to determine whether these spells are counted against your known / prepared spell limit.
What about sorcerers?
You may notice that the rules I quoted above mention the spell-casting function in particular, so you might wonder how they interact with sorcerers who use the pact magic function instead of the spell-casting function. On this topic, the rules for the multiclass spell checker only specify:
Pact spells. If you have both the Spellcasting class feature and the Pact Magic class function of the Warlock class, you can use the spell slots gained through the Pact Magic feature to cast spells you know or prepare from classes using the Spellcasting class feature You can cast known sorcerer spells on the spell slots you receive with the Spellcasting class feature.
Technically speaking, it is not mentioned how well-known spells are determined for sorcerers who are multi-classified with another class, but only how multi-classes interact with their pact magic slots and which slots can be used to cast hexes. However, based on the given logic and the lack of rules to the contrary, it seems clear that their known sorcerer spells are determined as if they were single-level warlocks, just like any other spell class.
Rule designer Jeremy Crawford unofficially acknowledges that this is also the intention of the rules in a series of tweets in December 2017:
The Cleric Spell feature states that you can cast spells up to your spell slot. Pact Magic gives you higher tier magic slots. Can a 1/5 cleric prepare an animated death? Because Warlock does not have a spell function, it does not use preprocessing rules for multiple classes
Spellchecker for multiple classes. You choose the spells you know / prepare for each class individually and pretend that you have only that class. The slots you receive from the multiclassing rules do not apply. You have 1 level in clerics? They prepare cleric spells as first-level clerics.
"If you have multiple classes, but the magic feature is only available for one class, follow the rules described in this class." Warlocks do not grant spellcasting, so she would follow the clerics. "Spells must have a level for which you have spells." Is that still wrong?
Continue reading and start with the next sentence: "Spells known and prepared. …"
"If you have the magic function of more than one class, apply the following rules." In this case, I would not have the magic function of more than one class, so the following rules for known and prepared spells would not apply.
Keep going until the end of the page. You will be redirected to the "Pact Magic" section, which explains how Pact Magic interacts with the spell. It does not affect what you can prepare.
The spell function does not mention where the spell fields originate from.
After rereading the rules for several classes, we turn to the cleric. The first sentence of "Preparing and Casting Spells" reads: "The cleric table shows how many spell bracelets you must use to cast your spells …" This original text forms the basis for what follows in this rule.
"The spells must have a level for which you have spells." However, the text in Preparing and Casting Spells does not explicitly state that it applies only to Cleric Spellworks. I do not feel that the first line restricts you this way.
When I wrote this rule, I started it with this sentence for a reason: putting everything you read about it in that rule into context. Phrases in our rules should not be interpreted in isolation from each other.
Thus, it is clear that determining known / prepared spells should work the same way in a sorcerer as in any other combination of spell classes.
Back to example:
How do I determine which spells I know / prepare, for example when I play a character in several classes with 1 level in the Warlock, 2 levels in the Bard and 3 levels in the Paladin?
We need to look at the magic functions and the class table of each class to see how many spells are known or prepared for each class:
The Warlock table shows that a first-level Warlock knows 2 Cantrips and 2 more spells. And according to the Pact Magic feature, "In the first level, you know two first-level spells of your choice from the Warlock spell list." (If you learn a new spell or swap an existing spell in higher Warlock levels for a new one, your new spell does not have to be higher than your Pact magic slots.)
Per Bard table, a 2nd level Bard knows 2 cantrips and 5 more spells. Like the sorcerer's pact magic feature, the bard's spell-casting ability states that your non-cantrip spells must be a level for which you have spell slots. As the Bard table shows, a 2nd level bard has only 1st level spells, so all 5 spells must be 1st level spells.
Finally, paladins prepare their spells instead of knowing / learning them. Paladin's Spellcasting feature states that you can "prepare a set of Paladin spells that match your Charisma modifier + half of your Paladin level, rounded down (at least one spell)." It is added that the spells must have a level for which you have spell slots. According to the Paladin Table, a third-tier paladin has only first-tier spells. For example, your paladin has a charisma value of 16 and a charisma modifier of +3. This means that you can prepare a number of spells equal to your Charisma Mod (3) plus half of your Paladin Level (3/2, rounded to 1) for a total of 4 spells.
That's not all. In the third level Paladins receive the function "Holy Oath". Each oath grants them oath spells in predetermined levels. Your Oath spells are always automatically prepared and do not count toward the number of Paladin spells prepared (and always counted as Paladin spells). The oath of surrender Paladin's Eid spells at Paladin level 3 are Protection from good and evil and sanctuary, This raises your actual number of prepared Paladin spells 6.
As Warlock 1 / Bard 2 / Paladin 3 with a Charisma Score of 16:
- Know 2 Warlock Cantrips and 2 Level 1 Witchcraft spells
- Get to know 2 bard cantrips and 5 first-level bard spells and
- Have prepared 4 Paladin spells of the first level and 2 always prepared Oaths
Spells specified in the subclass description (e.g. protection against
bad and good and sanctuary for a devotional paladin)
You would Not You have access to 2nd level or higher spells in any of these classes, unless you have at least 3 total sorcerer or bard levels, or at least 5 total paladin levels.