Let's review the basic numbers first, before we get into how Wizards break them: at level 1, every character picks 2 at-wills, 1 encounter power, and 1 daily. At level 2, they pick up a utility power too.

At-will powers

So, you have to choose two at-wills, right?

But there's something else first. Some classes have feature powers, and the Wizard's are their four cantrips. You get these automatically. You already have four at-wills, just by being a Wizard.

These do not count toward the 2 you pick. So pick two more Wizard Attack 1 at-wills.

Notice that the Cantrips have no level mentioned. Neither do Racial powers. This is your hint that it's an automatic feature that you get in addition to however many at-wills/encounters/dailies/utilities you get at your level. (The Paladin also has four class feature powers, but two of those are encounter powers.)

Total level 1 at-wills: 6. And then some racial features might give you more.

Encounter powers

1 at first level. Gain encounter powers the same as any other class.

A lot of races also give you an encounter power, but like I mentioned about the at-wills: this comes in addition to whatever single encounter power you pick, and doesn't count toward your limit of 1.

The Spellbook: Dailies and Utilities

This is where Wizards have some novelty, and where they get a bit complicated.

Every class has a notion that they have X powers and they have them available every day. Wizards break this mold and do things very differently.

When the Wizard might learn a daily or utility power — including their first-level daily and second-level utility — instead, the Wizard picks two dailies, or two utilities. They write these two down in their spellbook. Then as the feature says...

After an extended rest, you can prepare a number of daily and utility spells according to what you can cast per day for your level. You can’t prepare the same spell twice.

You pick a bunch, and these are your daily and utility powers for the day. So you know twice as many daily powers and twice as many utility powers as any other class. But at the beginning of your adventuring day, you pick the same amount as anyone else, and those are your dailies and utilities for the day.

So at level 2, instead of the usual 1 daily and 1 utility, you'll have twice that many - 2 dailies and 2 utility powers - written in your spellbook. But after you rest, you'll pick 1 of those dailies and 1 of those utility powers, and you'll have those available to use for the day. So on any given day, you still have access to the same amount of daily and utility powers as any other character - but uniquely, Wizards can change the powers they have access to day-to-day between those in their spellbook.

At a level where you might be alotted 3 daily powers, you'll have 6 in your spellbook, and pick any 3 of those 6 at the beginning of an adventuring day. Etc.

Rituals

You also get to start with some rituals written in your spellbook. This is separate to your powers entirely, and a different casting system every player character shares. You just get to start off knowing some rituals, where others will have to find them or buy them.

Yes, the wording does mean that a cleric of the Light Domain gains 'Fireball' as part of the Cleric Spell List; and here's why the wording is different. I hope the following helps explain why

"The following spells are added to the warlock spell list for you," which shows that Fiend patron warlocks have Fireball in their spell lists, and so should be able to use Spell Scrolls of fireball.

The above Warlock phrasing, just as it's phrased for Wizards and other spell casting classes that gain extra spells through means of specialization, is phrased that way because each of those classes have a spell list that they must choose spells from as they level up. They do not inherently know all spells available to a Warlock. So the phrasing in this case means the fireball spell is added to the list of spells he can POTENTIALLY add to his 'Spells Known' list. For intents and purposes, those are now on his Warlock Spell List

Now enter Cleric, and the Paladin; which have very different methods of spell learning.

If you have a domain spell that doesn't appear on the cleric spell list, the spell is nonetheless a cleric spell for you."

It must phrase it like this because Clerics and Paladins do not have to choose what spells they learn as they level. Clerics and Paladins inherently have the knowledge of every spell on their list....because we're awesome like that....and can prepare spells as needed in between long rests. If I wanted to go with cure wounds one day and then give up cure wounds to take Inflict Wounds after a long rest? I can do that freely. I have direct access to every spell available to Clerics. So when I choose Light Domain and it says the Fireball spell that isn't on my class list nonetheless becomes a cleric spell? It's now part of my knowledge of cleric spells (On my list of spells), and thus able to be utilized in scroll form since I now have the knowledge of how 'Fireball' works magically.

The only reason it doesn't say these spells are added to your spell list for you, is because it wouldn't matter in a Cleric or Paladin's case. Any spell that's on our spell list is a spell we can cast should we choose to. Any spell that's on a Warlock/Wizard/Sorcerer spell list, is a spell they can cast only if they chose it from their spell list when gaining levels.