How exactly do I obtain new spells?

[PozerWolf]

Using Gale, he's hitting level 3 and I select for him to learn 2 new spells. One is Hold Person, yet it doesn't appear on my spells list. Not wearing any gear that'll prevent this from appearing, even took a Long Rest and my new spells don't appear.
What is it that I'm doing wrong?

[tet5uo]

You can open the spellbook and "prepare" spells to use in combat. They can't all be prepared at once. You can also learn new spells with scrolls on the Wizards in exchange for gold.

[Dagný]

go to your inventory and there is a tab called spells there, look here

https://ibb.co/nzN83Yv

cantrips can be used once per turn
the ,,infinity" ones are always prepared
just below that one is the one you have to prepare, view it as equip spells, so the spells to the right of that spellbook are the ones I can use in battle

III
II
I

are the levels, I use 2 lv III spells (blue blocks)
3 lv II etc, but I can only equip 8 spells

[Nuee]

You also cannot learn spells from scrolls until you have reached the level of that scroll.

[PozerWolf]

Ah, I see now thank you very much!
And for references; the hotkey to access the spellbook is K.

[THE AVERAGE]

Just an FYI for those that didn't realize this, or for a future playthrough: one level of Wizard allows any caster to scribe any scroll up to their caster level. Caster level, not Wizard level. Your caster level is a combination of all levels in all caster classes you currently are, and some classes are considered "full" casters (like Cleric, Wizard, Sorcerer), some half, and some third.

You will only unlock four additional memorization slots, but you will still have access to ALL of the spells from the Wizard list, which is generally the most robust control/damage/utility list as you find the scrolls for them. Those memorization slots are generic too for the Wizard, which means they can be a mix of ANY spell from ANY level you currently know from the Wizard.

If you are not playing an INT character, simply pick all of the spells that do not care about that, like Haste, Misty Step, Dimensional Door, Greater Invisibility etc, and swap your four memory slots as needed. Taking 1 level of Wizard also allows you to refresh 1 level 1 spell slot per long rest, and since "spell slots" are generic and universal (besides the Warlock specific ones), you can refresh *any* level 1 spell.

For for a Sorcerer, it's a good idea to take 1 level of Wizard once you reach level 7. This lets you get your biggest early power boost at level 5, and then your subclass stuff at level 6.