The chart for abilities and spellcasters can also be found here
. It shows the bonus for every spell level except zero (as one doesn't receive bonus zero level spells).
Whether or not spells can be learned depends on the class. For some classes (bards, sorcerers), they have a fixed number of spells they know. The only way they can learn more spells is by taking a feat. There are some objects that can let them act as though they knew more spells, but nothing that actually grants them more spells known. Some classes (druids, clerics, rangers) know every
spell on their list. Most often these classes prepare spells, and choose from anything on their class spell list. Finally, wizards (and archivists) can learn any new spell they like, provided they can find a scroll or another spellbook to copy it out of. This costs time and money (but not experience).
So, let's go with a few examples. Rainbow Unicorn the druid
is level three. Her base spells per day are 4 zero level spells, 2 first level spells, and 1 second level spell. She has a wisdom score of sixteen. This gives her one bonus spell at both first and second level. She would also receive a bonus third level spell, but she can't cast those yet. So, a total of three first level spells and two second level spells. She can prepare any spell on the druid spell list. So, at level zero, she prepares detect poison, know direction, and resistance twice. At level one, she prepares goodberry, jump, and speak with animals. At level two, she prepares hold animal and owl's wisdom.
Vornax the sorcerer
is level four. He has a charisma score of 18. His base spells per day is 6 0th, 6 1st, and 4 2nd level spells. With his charisma score, this gives him a bonus 1st and a bonus 2nd level spell (he'd also receive bonus third and fourth level spells, but again, he can't cast those yet). So, his spread is 6, 7, 5. Unlike Rainbow, he doesn't have to pick which spells he's going to use in those slots ahead of time. However, he only knows a small number of spells. 6 0th level spells, 3 1st level spells, and 1 second level spell. His 0th level spells are read magic, detect magic, resistance, ray of frost, ghost sound, and prestidigitation. For his first level spells, we'll say he knows magic missile, charm person, and sleep. His second level spell is scorching ray. He can cast any of these so long as he has the spell slots to do so. He can even use second level spell slots to cast first level spells (though he gains no benefit from doing so without metamagic feats).
Now, Omgwin Lolhax the wizard
is level five, and has an intelligence score of 20 (he has a +2 circlet of intellect). His base spells per day are 4 0th, 3 1st, 2 2nd, and 1 3rd. Going by the chart, his spread becomes 4 0th, 5 1st, 3 2nd and 2 3rd. Now, as I said before, wizards can learn any spell on their list if they spend the time and money. His base spells known are 3+int spells at first level, and then two more at each subsequent level. Plus all cantrips. So, excluding cantrips, he knows 14 spells. For convenience sake, we'll say he learned the highest spell he could at each level, making it 8 1st level spells, 4 2nd level spells, and 2 3rd level spells. He's also bought scrolls to copy from (and paid to copy from other wizards' spellbooks as well). So we'll say he actually has 12 1st level spells, 6 2nd level spells, and 3 3rd level spells. He prepares his spells from those in his book. We won't get into just what he has in his spellbook, but we'll say he has as his prepared spells as follows: 0th are detect magic, read magic twice, and daze, 1st are grease, mage armor twice, true strike, and color spray. 2nd are glitterdust, web, and alter self. Third are haste and ray of exhaustion. This is a lot oless convenient than the sorcerer's spontaneous spells, but the wizard can completely change out his spells prepared the next day and have a completely different strategy ready.
Hopefully that makes everything clear.