Static Armor You generate a shocking aura on a willing creature's armor that damages attackers.

Spectral Shield You invoke a minor protective force to grant yourself a +1 bonus to your AC.

Hydro Barrier Throw water in front of yourself to blunt an incoming attack.

Haptic Counter You focus on the energy kept within you and release it in a defensive manner.

Glyph Trap You cast a glowing pool of magic upon the ground by drawing a glyph with paint, or something that can mark a surface, that damages and hinders anyone who wanders through or is pushed into it.

Furious Charge You create a shield of force on your shoulder and ram it into your foes.

Elemental Aegis You protect a creature with a subtle magic ward.

Disjunction You channel a sliver of magical energy into your targetвЂ™s cantrip as it forms, attempting to destabilize the casting.

Astral Barrier Reinforce yourself with starstuff to blunt an incoming attack.

Arcane Deflection With a wave of your hand, you create a barrier of arcane energy to deflect attacks.

(These can be any combination of spells he knows, such as 4 first level spells and 4 second level spells even though he only has 3 second level spell slots available. At level 4, let's assume the same wizard increases his Intelligence to 18 (+4 modifier): he can prepare 4 + 4 = 8 spells out of the 6 + (2 x 3) = 12 he knows total. So for example, let's say your wizard has 16 Intelligence (+3 modifier) at level 1: he can prepare 1 + 3 = 4 spells out of the 6 he knows total. Therefore, each level you gain allows you to prepare at least one additional spell, but possibly two if you also increase your Intelligence Modifier. You have the flexibility of changing this set of spells each day, but you are limited to a small subset of the spells you know, that number is: Intelligence Modifier + Wizard level.

(So a level 2 wizard learns an extra two first level spells, but a level 3 wizard can learn two more first level spells, two second level spells, or one of each.) Plus, you could potentially find spell scrolls or other wizards' spellbooks and learn new spells from those sources.ДЇinally, we have prepared spells, which are the only spells you can cast on a specific day. Each time you level up, you learn an additional 2 spells of any level for which you have spell slots.

Wizards start knowing 6 spells at level 1 (which are all first level spells because that's the only spell slot available to you). Spells known is also pretty easy, but it's easy to miss when reading the class features. They can get more complicated if you multiclass, but that's a topic for another post (FYI, multiclassing also affects what spells you can learn). Spell slots are easy because they're given on the wizard class table. So there are three important things you have to keep track of as a wizard: the spells you know, the spells you have prepared, and the number of spell slots you have available. To answer your final question, 18 INT would indeed increase your number of prepared spells to 7, assuming you had 18 INT at 3rd level. Any combination is valid, as long as you have a slot with a level no less than the level of the spell you want to cast.

You could also prepare only one 1st level spell, use all of the 1st level slots for that spell, and decide during the adventuring day which of your five 2nd level spells should be cast using your two 2nd level slots. You could prepare six different 1st level spells and just cast some of them using your 2nd level slots.

However they still can only prepare a fixed number (wizard level + INT mod) of spells each day. There's no upper limit to how many spells a wizard can have in their spellbook(s), which is part of what makes them such versatile casters. The spellbook contains the actual spells the wizard knows, starting with 6 at 1st level and gaining 2 more with each additional level. This wizard, assuming they have 16 INT, can prepare 6 (3+3) spells from their spellbook. Spell slots are your resource for casting spells. The 3rd level wizard does indeed have four Lv 1 slots and two Lv 2 slots but that only determines how many times the wizard can cast their prepared spells between rests. You're conflating the terms 'spell' and 'spell slot' which is the source of your confusion.

5e wizard spells