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I have only ever played Pathfinder and the way it works in that system is you would get extra spell slots based on how much intelligence you had. So in this table, it shows how many more spells per day you get based upon how much more intelligence you have.

Do you get any extra spells as a spellcaster in D&D 5th edition?

This is all I could find in the D&D 5th Edition players handbook:

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.

Nothing says anything about bonus spells slots. I just want to confirm that you do not get bonus spells in 5th edition D&D, right? Because if the chart on page 113 of the PHB says to only choose two 1st level spells to prepare as a 1st-level wizard, what do I follow?

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Okay, so, let's start with:

Forget (almost) everything that you know about Vancian Magic from Pathfinder (or 3.5e)

As you mentioned, you don't get bonus spell slots OR bonus spells from high int.

In your question, you mention

Because if the chart says only choose 2 1st level spells to prepare in this chart

And this statement is where you are misunderstanding the concept of prepared spells in 5e. Unlike Vancian Magic, you can prepare more spells than you can actually cast. You also only need to prepare each spell once (and you can cast it more than once).

So, we have two concepts here. The first is prepared spells, which are given by Wizard Level + Int Modifier. The spells you have prepared are the spells which you can cast during the day. The second concept is spell slots. Your spell slots tell you how many spells of that level you can cast during the day.

Trying to simplify it:

  1. Prepared spells answer the question "What spells can I cast?"
  2. Spell slots answer the question "How many spells (from each level) can I cast?"

Again, notice that these two concepts are not exactly related. You can cast as many prepared spells as your spell slots let you, without having to prepare them more than once or anything like that. You can also prepare more (or less, if you are playing a Dumb Wizard) spells than you can actually cast during the day.

TL;DR: Spell slots are not the same as prepared spells in 5e.

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    \$\begingroup\$ So... Prepared spells- What spells am I capable of casting? Spell slots per level - How many times I can cast a spell. Is this correct? So everyone is like a modified sorcerer from Pathfinder? \$\endgroup\$ – Philip Sturchio IV May 14 '18 at 0:34
  • \$\begingroup\$ @PhilipSturchioIV Coincidentally, that is exactly the edit I was making while you commented. With almost the exact wording, lol. \$\endgroup\$ – HellSaint May 14 '18 at 0:36
  • \$\begingroup\$ @PhilipSturchioIV Yes. Everyone is a lot more like a Sorcerer than the Vancian Wizards from 3.5/Pathfinder. \$\endgroup\$ – HellSaint May 14 '18 at 0:38
  • \$\begingroup\$ @PhilipSturchioIV Not sorcerer, wizard in 5e still seem to have unlimited spell known, so more like Arcanist: d20pfsrd.com/classes/hybrid-classes/arcanist \$\endgroup\$ – Nam Nguyen Hoang May 14 '18 at 3:40
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    \$\begingroup\$ @PhilipSturchioIV I think the "what/how may" split is the key here: unlike 3.x/PF, you don't prep "two fireball, one grease, and a detect magic" in 5e - you prep "fireball, grease, detect magic", and then you have "4 first-level slots and 2 second-level slots". Any of those slots can cast any of those spells as long as the level of the slot is equal or higher to the level of the spell. Also worth noting: you can cast cantrips as many times as you like; they're unlimited-use and you don't prep them, you just learn them when you level up and then they're always available. \$\endgroup\$ – anaximander May 14 '18 at 8:23
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Correct, you do not gain bonus spell slots from high Intelligence

There are other means of gaining bonus spell slots from class abilities and magic items, but high Intelligence does not grant any bonus spell slots in 5e.

As you have pointed out, it does grant additional prepared spells instead. These are spells the Wizard can cast before resting using their spell slots. The more prepared spells a Wizard has, the more versatile a Wizard can be during encounters.

The number of spell slots the Wizard has limits how many spells they can cast before resting. For a 1st level Wizard, they have 2 1st level spell slots with which to cast their [1 + Intelligence modifier] prepared spells (4 prepared spells at Int 16, for example).

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  • \$\begingroup\$ So to get this straight. You do not learn bonus spells. However, If you are level 1 and are allowed 2 first level spells by default, if you have 22 Intelligence (Modifier of of +6) How many spell slots do you have in total? Do you add 7 spell slots because of your level and ability modifers? \$\endgroup\$ – Philip Sturchio IV May 14 '18 at 0:28
  • \$\begingroup\$ @PhilipSturchioIV You are confusing spell slots with prepared spells. It's common since earlier editions (3.5e and Pathfinder, which follows most of 3.5) used the concept of Vancian Magic. Fifth edition doesn't. The number of spells you can cast (spell slots) during the day doesn't change with your intelligence score. \$\endgroup\$ – HellSaint May 14 '18 at 0:31
  • \$\begingroup\$ @Philip Sturchio IV Edited answer to clear up any confusion between prepared spells and spell slots, hopefully. \$\endgroup\$ – Tenryu May 14 '18 at 1:36
  • \$\begingroup\$ But to answer the question (however a level 1 character got such a powerful magic effect), a level one Wizard with 22 Int can after each long rest prepare 1 (wiz level) + 6 (Int bonus) = 7 different level 1 wizard spells. He/she still only has two spell slots, so gets to choose two of them to actually cast, or the same one twice. Once a day, a short rest can restore one expended spell slot, allowing a third casting. \$\endgroup\$ – aschepler May 14 '18 at 11:50
  • \$\begingroup\$ @aschepler Question has changed- it no longer references 22 Int. My answer does specify spell slots limit casting before resting so it accounts for Arcane Recovery. \$\endgroup\$ – Tenryu May 14 '18 at 13:40

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