Wizards learn new spells when they gain a level.

Learning Spells of 1st Level and Higher

Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook for free. Each of these spells must be of a level for which you have spell slots, as shown on the Wizard table. On your adventures, you might find other spells that you can add to your spellbook (see the “Your Spellbook” sidebar).

If your campaign starts with new characters above level 1, what number of spells might the typical spellbook for a wizard contain? Using the RAW indicates a minimum number. However, a wizard adventuring for a few levels before your campaign starts could discover scrolls, meet other wizards to trade spells, find a spellbook as treasure, and earn sufficient gold to pay the cost of copying the spell. 

This is entirely up to the DM. Spell scrolls are magic items and have rarities for the level of spell they might have and the Starting Above 1st Level section has recommendations of starting magic items and gold by level, so a DM could default to that.

I use the Starting At Higher Levels chart (https://www.dndbeyond.com/sources/dmg/creating-a-campaign#StartingatHigherLevel) and figure it out based on how much gold they get compared to the cost to scribe, but you also gotta factor in the cost of any spell components for each spell too. That’s probably about as close to a RAW answer you’re likely to find.

The chart that shows how many spells a Wizard automatically learns is in the PHB, and in the free Basic Rules: (https://www.dndbeyond.com/sources/basic-rules/classes#Wizard).

There’s also this handy dandy guide on DM’s Guild a friend of mine made: (https://www.dmsguild.com/m/product/417147).

The chart in the DM’s Guide that I mentioned shows how much money a character starts with if they start at a level higher than 1st.

Quote from IamSposta >>

There’s also this handy dandy guide on DM’s Guild a friend of mine made: (https://www.dmsguild.com/m/product/417147) 

This chart is nice when you want to learn how many spells you learn free when you gain a level - and how you might allocate them. The chart reflects the minimum the wizard learns and knows - cumulative learning of 2 spells per level. The DM Guild chart helps establish the available funds for spell scribing - helpful too.

Any estimates of how many spells the wizard acquires at each level in their adventures and scribes into their spellbook? Perhaps one estimate is to add 50% - 150% of the number of spells they learn for free. For example, a wizard at level 1 begins with six spells in their spellbook and might acquire 3 to 9 additional spells to scribe in their spellbook while they gain experience to attain level 2.

As I wrote in one of my previous posts, I base it off of how much money the Wizard would have at their level based on the chart in the DMG:

Quote from IamSposta >>

I use the Starting At Higher Levels chart (https://www.dndbeyond.com/sources/dmg/creating-a-campaign#StartingatHigherLevel) and figure it out based on how much gold they get compared to the cost to scribe, but you also gotta factor in the cost of any spell components for each spell too. That’s probably about as close to a RAW answer you’re likely to find.

 
So, for example, a 5th—10th level Wizard would have 500 gp + (1d10 × 25 gp). I would guesstimate that a 5th level Wiz would have 525-550 gp, and a 10th level Wiz would have 725-750 gp.

Now, a 5th level Wiz could cast spells up to 3rd-level, and the spell scribing rule for the Wiz class states the following:

For each level of the spell, the process takes 2 hours and costs 50 gp. The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it. Once you have spent this time and money, you can prepare the spell just like your other spells.

So copying a 1st-level spell costs 50 gp to scribe, an 2nd-level spell costs 100 gp, and a 3rd-level spell would cost 150 gp. So if a 5th level Wis has 550 gp, then they could add at most 11 1st-level spells, 5 2nd-level spells with 50 gp left over, 3 3rd-level spells with 100 gp left over, or some mix of 1st- through 3rd-level spells. 

So, for example, maybe the Wiz has 1 additional 3rd-level spell (150 gp), 2 additional 2nd-level spells (200 gp), and 4 additional 1st-level spells (200 gp) for a total of 550 gp.

Now, if the spell has any M components that cost money (like identify), that cost has to get added to the cost to scribe and must fit within that 550 gp budget.

Make sense?

Correct, those copying spells rules only applies to spells they collect outside of regular character progression.