This is my first time multiclassing a character.

I wanted to multiclass my bard with something, and I wanted to make sure I have the rules for multiclassing correct - because with my understanding, there's no way what I'm about to say could make sense.

Important Stats: Charisma: 19 (+4), Intelligence: 19 (+4)

My 6th-level College of Lore Bard knows 9 spells. My feature Additional Magical Secrets gives me 2 more spells. If I read the multiclassing rules correctly, then by multiclassing into wizard, my character can then prepare 5 more spells out of his 6 from his spellbook.

So in total: 9 Known + 2 + 5 Wizard = 16 Spells he knows at once. Is this right? For a level 7 character, that seems like a lot.

(Side note: I understand the number of spell slots I can use.)

Current spells known: 11 bard spells, 6 Wizard spells

A 6th level bard typically knows 9 spells, but the College of Lore lets you steal an additional 2 spells at 6th level. That gives you 11 bard spells known.

A 1st level wizard gives you 6 first level spells, so that's an additional 6 spells.

You will also know a total of 3 bard cantrips and 3 wizard cantrips.

Your 1st level wizard can scribe more spells

Also important to note that if your wizard is able to find scrolls, they will be able to scribe them into their spell book to increase their options.

When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a spell level you can prepare and if you can spare the time to decipher and copy it...

For each level of the spell, the process takes 2 hours and costs 50 gp.

Spells Prepared: 11 bard, 5 wizard

Your bard will always have their spells prepared, but the wizard is limited by their INT modifier. Since your modifier is +4 (from INT 19), you can prepare 5 of the 6 spells you know.

That gives you your 11 bard spells, 5 wizard spells, and 6 cantrips ready to be used each day.

Casting

Do note that your bard spells will utilize your CHA, while anything cast from your wizard list will be based on your INT.

Multiclassing rules also define your total spell slots for casting at (4) 1st level spells, (3) 2nd level spells, (3) 3rd level spells, and (1) 4th level spell.

Also note that you are limited by your spellclassing cast as to what level you can cast:

If you have more than one spellcasting class, this table might give you spell slots of a level that is higher than the spells you know or can prepare. You can use those slots, but only to cast your lower-level spells.

An INT 19 Bard 6/Wizard 1 would have 16 spells available to cast

You've got it right. The relevant rules that back you up are:

• The Bard Table (PHB, p. 53) shows that you have 9 spells known at level 6.


• 
  

  As a College of Lore Bard, you have Additional Magical Secrets:

  
  
  

  
  

  At 6th level, you learn two spells of your choice from any class. A spell you choose must be of a level you can cast, as shown on the Bard table, or a cantrip. The chosen spells count as bard spells for you but don’t count against the number of bard spells you know.

  
  
  

  -- Additional Magical Secrets; PHB, p. 55

  
  

  
  


• 
  

  The Spellcasting rules for a Wizard show that:

  
  
  

  
  

  You prepare the list of wizard spells that are available for you to cast. To do so, choose a number of wizard spells from your spellbook equal to your Intelligence modifier + your wizard level (minimum of one spell). The spells must be of a level for which you have spell slots.

  
  
  

  -- Preparing and Casting Spells; PHB, pg. 114.

  
  

  
  

And you would have a total of 17 spells known

In addition to the 11 spells known from being a 6th level College of Lore Bard, the Wizard's Spellcasting feature also says this about spells known:

At 1st level, you have a spellbook containing six 1st-level wizard spells of your choice. Your spellbook is the repository of the wizard spells you know, except your cantrips, which are fixed in your mind.

-- Spellbook; PHB, p. 114

This would give you a total of 17 spells known overall (as you've already noted, only 5 of those 6 could be prepared at once with your current INT modifier).

Isn't that a lot for a level 7 character?

Generally, you can get more spells as a multiclassed character than you could as a single class character, and remember that your Additional Magical Secrets is one of the perks of being a Lore bard, multiclassing or no. Lore bards are supposed to know a few more spells than other bards. That said, while multiclassed casters may have more spells than a single-classed caster, their spells are also necessarily lower-level than a single-classed caster of the same character level could have.

Spells known vs Spells prepared

You may be confusing the concepts of spells known and prepared.

Bards have a number spells known. From the Spellcasting section for Bards:

Spell Slots

The Bard table shows how many spell slots you have to cast your bard spells of 1st level and higher.

(...)

Spells Known of 1st Level and Higher

You know four 1st-level spells of your choice from the bard spell list.

The Spells Known column of the Bard table shows when you learn more bard spells of your choice.

This means that any spell you know as a Bard is always available to cast.

The wizard has both a spellbook and a number of spells prepared:

Spellbook

At 1st level, you have a spellbook containing six 1st-level wizard spells of your choice. Your spellbook is the repository of the wizard spells you know, except your cantrips, which are fixed in your mind.

Preparing and Casting Spells

The Wizard table shows how many spell slots you have to cast your wizard spells of 1st level and higher. (...)

You prepare the list of wizard spells that are available for you to cast. To do so, choose a number of wizard spells from your spellbook equal to your Intelligence modifier + your wizard level (minimum of one spell).

This means wizards store all their spells (other than cantrips) in a spellbook, then choose a number of them each day to have prepared, which are the spells that will be available to cast using spell slots for that day. There are methods to copy spells into a wizard's spellbook which means that they can have a nearly unlimited number of spells known but they can still only prepare a few of them each day.

In your case, the spells that you have available to cast are all spells known as a Bard plus any spell prepared as a Wizard, in which case the total number of spells available is indeed 11 Bard spells plus 5 Wizard spells but you actually know 11 Bard spells and 6 wizard spells (or more if you add some to the spellbook)

The high number of available spells is not an issue

Yes you will have a total of 16 spells available to cast, which may seem a lot for a 7th level character, but this is merely a side effect of multiclassing.

When multiclassing spellcasting classes you tend to get a lot of different spells with the downside that you get less higher level spells (remember that the maximum level for spells learned depends on individual class levels), regardless of available spell slots.

For example: with your charisma of 19 and Intelligence of 19, you could also have these levels at total character level 7:

• Bard 2 (4 spells),


• Sorcerer 1 (2 spells),


• Warlock 1 (2 spells),


• Paladin 2 (5 spells prepared),


• Wizard 1 (5 spells prepared)

For a total of 18 available spells! If you have the stats to also multiclass into cleric or druid you could even increase this number further. But do note that in my example all of these spells are lvl 1 spells which may not be ideal.