The method can vary from class to class. Wizards can do what you describe, adding new wizard spells to their spellbook, but most spontaneous casters (such as bards or sorcerers) don't get that option.
Still other classes, such as clerics and druids, know ALL of their class spells just as soon as they are able to cast them.
I'm not sure what you have included in "casters", but wizards, given the time and money and access and passing the checks, can surely copy any spell they come across into their spell books. There is no maximum number of spells a wizard can have in her books.
If you're referring to the table that includes their class features, base attack bonus, and saves, then you're still incorrect--the values there represent the number of spells the wizard can prepare (and therefore cast) in a single day.
So, where it says Spells Per Day--1st--1, that's one spell the wizard can prepare to cast each day at level 1. He can prepare any spell of that spell level (or a lower level) that he has in his spellbook and cast it one time until he gets a full night's sleep. A wizard may have quite a large number of spells available to select from.
Note, also, that a Specialist wizard gets 1 additional spell to prepare/cast per day at each spell level, but it has to be a spell from his chosen school.
Additionally, casters gain additional spells to cast each day based on their casting stat (the table for this is earlier in the book), which will typically mean 1 additional spell per level for at least the first few spell levels.
All totaled, your average level 1 PC wizard will have 1 (class) + 1 (specialization) + 1 (bonus from a high Int), or 3 level 1 spells to cast each day, to be selected from all the spells in his spellbook.
0 level spells (cantrips), work a little differently, in that the number in their column represents how many you can prepare each day, but they may be cast as often as you like. Finally, 0 level spells benefit from neither school specialization, nor high stat bonus spells, so the number available doesn't change the way other spells do.
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Table 1-3 on page 17 shows bonus spells per day due to high casting stat
Table 3-14 on page 72 shows spells per day (sorcerer)
Table 3-15 on page 73 shows spells known (sorcerer)
Table 3-16 on page 80 shows spells per day (wizard)
Prepared casters either know their whole list (cleric, druid, paladin, ranger) or can learn it (wizard). Spontaneous casters (bard, sorcerer) only learn spells by leveling up. They get more castings in return for fewer choices. Some FCB (favored class benefit) choices let spontaneous casters learn more spells. That changes the balance greatly.
This pattern continues for classes in other books, and for prestige classes.
/cevah
yes, I was reading that table (using your example) to mean that he could prepare 3 spells per day, but could also only have 3 spells in his book, so if he prepared the same spell 3 times, his book would have the other 2 he knows, didn't realize that there could be 20 others that he knows.
Yup. And he can prep them in any combination, though it looks like you've got that part down.
And as Cevah points out, my previous post doesn't really hold true for sorcerers and other spontaneous casters--their similar table indicates spells per day in the same way, but they also come with a "Spells Known" table, and that works more like how you were initially thinking.
If a sorcerer has 2 spells known, and can cast 4 spells per day of that level, then he can cast a total of 4 spells in the day of any combination of those two he knows, and the known spells are difficult to increase outside of leveling (some favored class bonuses will grant additional spells, and there are some magic items that allow you to add a spell to your list of known spells while you're holding it).
A high casting stat (Cha in the case of the sorcerer) works the same, increasing the number of castings per day.
Finally, you have a third group of casters that work like the wizard, but who have access to ALL spells on their class' list when preparing spells, without being limited by those learned by leveling or having to scribe anything. This is how most divine casters (cleric, druid, paladin, ranger) function. They still can prepare only a small number of spells each day, as indicated on their respective tables, but they have many more options to stick in those slots.
ok, that makes sense now, I was thinking all casters worked the same, which was the root of my confusion. Thanks for clearing this up.
You could broadly separate casters into 3 groups: prepared off of a limited, but expandable spell list (wizards and witches); prepared off of an unlimited spell list (clerics, druids, paladins, rangers); spontaneous off of a limited list of spells known (sorcerers, oracles).
They all have their spells per day, but how they get those spells--what they have access to--differs from one group to the next.
