Yes you get to learn all those spells you mentioned, but magic always comes at a price. In this case while you "know" a bunch of spells you can still only cast a specific number of them each day. So at level 2 you can cast your at-wills 1 encounter, 1 utility and 1 daily spell (not including cantrips and rituals). You just get to choose which one you will, for lack of a better term, "charge up" for the day. Like Peteris said, at each extended rest you must choose which of your spells you will prepare for the following in game day. This keeps you at the same power level as other same level characters (except you get +1 at will that you can use at any time).

So, Bob the level 2 wizard opens his spell book in the morning after getting a good night sleep and turns to his "Utility Spells" page. He sees Expeditious Retreat and Shield, not wanting to get too close to enemies, he chooses to memorize the evasive spell Expeditious Retreat absorbing the spells magic for use later that day. He then flips through his spell book and does the same thing for the daily spells he has in his book. He then sets off on his travels with his companions being able to cast 3 at wills, 1 encounter, 1 utility and 1 daily.

The benefit here is if you know you'll be facing a certain creature and you have a spell that will deal lots of damage to it, you can prepare specifically for that engagement.

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.