Proficiency Bonus - Which determines Spell Save DC, Proficient Skill Bonus, Proficient Attack Bonus, Proficient Save Bonus.
Cantrip effectivenes - Per Jeremy Crawford's twitter confirmation linked here, though Wizards forgot to make that explicit in the PHB so you have to go looking for the confirmation. This said, if your primary class is a martial class, weapons are likely to deal more reliable damage so you may prefer to use cantrips with utility purposes or to give a reliable ranged attack,
Things that improve based on Class Level:
Access to higher level spells on a class list
Special Class Features
Ability Score Improvements and Feats (usually at 4, 8, 12, 16, 19), This is why I typically do theoretical multi-class builds in increments of 4 (so my theoreticals are more likely to be 8/12 than 10/10 despite my using 10/10 as an example here)
Hit Points and available Hit Dice for Short-Rest Healing, each class has a different Hit Die
Things multiclassing can give you:
Armor, Weapon and Tool Profciencies - This means that a 19th Wizard/1st Fighter can wear Plate Armor and cast spells. Yes, this is possible. There is no longer any arcane spell failure. However, it is not particularly a good idea. A character that is primarily a wizard most likely lacks the necessary Strength to use Plate Armor without suffering speed penalties. Also if encumbrance rules are in effect they are likely to find themselves encumbered. By and large, while wizards CAN use plate armor now as long as they are proficient, there are usually better magical options than plate armor for increasing AC. Note, this also means long-bow wielding wizards. A lot of multiclass options provide Tool, Armor and Weapon Proficiencies.
Things multiclassing sometimes gives you:
Skill Proficiencies: Most multi-classing does NOT get you new skills, but some multi-classing does.
Things that multiclassing does not give you:
New proficient saving throws: if you start as a Sorcerer and change to a Fighter, you do not gain proficiency in Strength Saving Throws. You need to use a Feat to gain access to a new Saving Throw proficiency unless a class gives it to you (as Rogue and Monk do).
Things that do not overlap from multiclassing:
Unarmored Defense - A barbarian/monk does not add Con and Wis to their AC, only the best of the two. Also, A draconic sorcerer/monk does not have AC 13 + Wis + Dex. They have the best of 13 + Dex or 10 + Wis + Dex.
Extra Attack - If you have Extra Attack from Ranger and Paladin, it does not stack to grant you 3 attacks. If you want three attacks, you need to take Fighter levels to the appropriate level.
Channel Divinity - a Cleric/Paladin gains new Channel Divinity options but does not gain new uses/rest.
How multi-classes spell-casting works:
You take your levels in each Full Caster class (Bard, Cleric, Druid, Sorcerer, Wizard) plus half your levels in each Half-Caster class (Ranger, Paladin) Plus one third your level in Third-Caster classes (Artificer, Eldritch Knight, Arcane Trickster). This creates your total caster level for determining how many slots you get.
A 10 Cleric/10 Paladin would have the spell slots of a 15th level Full Caster (4 1st 3 2nd 3 3rd 3 4th 2 5th 1 6th 1 7th 1 8th)
A 3 Bard/4 Ranger/3 Paladin/ 3 Eldritch Knight/6 Artificer/1 Cleric would have the spells slots of 10th Level Full Caster. (Paladin 3/2 = 1.5 rounded down to 1) or (4 1st 3 2nd 3 3rd 3 4th 2 5th)
However, the spellcaster would not have access to higher level spells than they would for their class.
The 10 Cleric/10 Paladin would have access to 5th level Cleric spells and 3rd Level Paladin spells.
They cast ALL of their spells from the same pool of spell slots, which is why some Paladin-only spells have wording related to being used with a spell slot of higher than 5th level which they wouldn't be able to gain without multi-classing.
The multiclass nightmare example would have access to 2nd Bard Spells, 1st Level Cleric Spells, 1st Level Paladin Spells, 1st Level Ranger Spells, 1st Level Eldritch Knight Spells, and 1st Level Arcane Trickster Spells. Which means that every spell slot they have from higher than 2nd Level would by necessity be devoted to over-powering low level spells.
Prepared casting would be by Class, a Wizard/Cleric would have two separate lists of prepared spells though both would use the same spell slots.
How Multiclassing Warlock works with Spellcasting
Warlock Pact Magic advances outside the Caster Spell Slot chart in terms of magic. In this case, figure out your other spellcasting abilities as normal and then figure out the number of Warlock Pact slots you get at that level as normal.
A Warlock 10/Wizard 10 has the spell slots of a Full Caster of level 10 (4 3 3 3 2) and the pact slots of a Warlock of level 10 (2 5th level slots). They would be able to cast Warlock spells with their Wizard slots and Wizard spells with their Warlock slots. This effectively they have the following Spell Slots (4 3 3 3 4)
However, I am not certain whether Invocations that require Warlock Spell Slots can use Wizard Spell Slots. so until I get/find confirmation on this I am going to assume that Invocations which consume Warlock Pact Slots can only be used with Warlock Pact Slots.
Also notice that the Pact Slots still recover on a short rest and the wizard slots still recover on a long rest.
Points to consider when Multi-classing:
Bonus and Reaction Usage - A lot of class features rely heavily on giving new uses for spending Bonus and Reaction usage. Having more options is always going to be useful, but be wary when two classes both give you essentially the same option. For example, Frenzied Berserker and Monk both allow the use of the Bonus Action to make attacks, though the Monk option also allows the use of a Bonus Action to make 2 bonus attacks by spending ki.
Can you get the flavor you want from a Feat? - There's a lot of dabbling that can be done just with the feats and they often make full multiclassing to create a desired concept unnecessary.
MAD - Multiple Ability Dependence - Ability Score Improvements are very thin on the field. Multiclassing can make them even more so. As such, be very wary of a build that's going to require you to have high stats in more than two Abilities.