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Each of these Spells must be of a level for which you have Spell Slots, as shown on the Wizard table. So, you get 2 spells every time you level up. But, you are limited to only spells that you have a spell slot for. Now, this may not be the most practical way to do this but, if a wizard were only to learn new spells of the highest spell slot ...
All those other classes' tables have a column headed "Spells known". The wizard table does not. It only has a column "Cantrips known". All these other classes have a subheading in their description titled "Spells Known of 1st Level and Higher", that describes the concept that they know a limited number of different spells. The wizard does not ...
See Spells Rules for the general rules of spellcasting and the Spells Listing for the wizard spell list. Cantrips. At 1st level, you know three cantrips of your choice from the wizard spell list. You learn additional wizard cantrips of your choice at higher levels, as shown in the Cantrips Known column of the Wizard table. Spellbook
Instead it tremendously increases the # of spells PREPARED (Known for classes besides Wizard/Artificer). An Artificer 11 can cast spells of up to 3rd level, and can prepare Int + level/2. For an Int of 18, that would be 9 spells. Plus 3 cantrips. An Artificer 6/Wizard 5 can learn Artificers spells of only 2nd level, but Wizard spells of upto ...
Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook. So, assuming you have a 16 Int, a third level Wizard can prepare 6 spells (3 int mod + 3 level), and your spellbook should contain 10 spells, probably 8 1st level and 2 2nd level.
1. For most spellcasters, you can find the number of spells known at the level in the class table, not the number of spells they learn. For example the sorcerer: The Spells Known column of the Sorcerer table shows when you learn more sorcerer spells of your choice. Each of these spells must be of a level for which you have spell slots.
Wizards don't really know spells (except for cantrips) — they add them to their spellbooks, which can happen while adventuring in addition to those they learn on level-headed. This is under "Learning Spells of 1st level or higher" in the book. Most "divine" casters — clerics, paladins, druids — don't learn individual spells .
There should be a button "Manage Spells" on the far right side. This brings up an extra panel that lets you add and remove spells based on your current class availability. The wonky one is Wizard, since their ability to enter spells into their spell book is determined more by the charitably of the DM than it is by character progression.
You might find other spells during your adventures. You could discover a spell recorded on a scroll in an evil wizard’s chest, for example, or in a dusty tome in an ancient library. Adding these scrolls to your spellbook costs 50 gp and 2 hours per level of the spell. (You can also copy spells you already know for 10gp and 1 hour per level.)
As another example, say you're creating a very high level Bard from scratch, starting at level 15. The book says your Spells Known is 19, and your spell slots are 1st × 4, 2nd × 3, 3rd × 3, 4th × 3, 5th × 2, 6th × 1, 7th × 1, and 8th × 1. So, for those 19 Spells Known, you can choose any spells from 1st through 8th spell level.