Do you improve your ability score at every level?

Question

Since I'm new to Dungeons & Dragons 5e, I don't understand how ability scores work. I searched the Basic rules and the Player's Handbook, but I can't find any clarification on this question: Do you gain 2 ability score points to add with every level gained?

I know there's something called Ability Score Improvement, and I think that means you get to improve 2 ability scores by 1 (or you can add both of them to the same ability) but is this in addition to the 2 you get every level?

I'm playing a level 3 Cleric, and I have a wisdom score of 17. My ally (level 3 fighter) has a strength score of 19 and a dexterity score of 18.

Can someone explain to me how increasing ability scores works?

Answer 1

No: you do not gain an Ability Score Improvement at each level.

I know there's something called Ability Score Improvement, and I think that means you get to improve 2 ability scores by 1 (or you can add both of them to the same ability) but is this in addition to the 2 you get every level?

There is no "2 ability points that you get every level." That rule does not exist.

Just like any other class feature, an Ability Score Improvement is gained only when it is listed in the Features portion of your class table. With two exceptions, classes in the Player's Handbook gain ASIs at Levels 4, 8, 12, 16, and 19. The Fighter also gains ASIs at Levels 6 and 14; the Rogue gains an additional ASI at Level 10. You can see these listed in each class's experience table.

When you gain an Ability Score Improvement (Basic Rules p. 22 / PHB. p. 59), you may either add +2 to one of your Ability Scores, or you may add +1 to two of your Ability Scores. This may not raise any Ability Score over 20.

As a note: ASIs are gained through individual classes, so if you multiclass into another class instead of continuing to Cleric 4, you will not gain an Ability Score Improvement until you either level up again and choose to advance to your 4th Cleric level or level up enough in your additional class to reach its first ASI at level 4.

Answer 2

While Reibello has already answered this question, it is also confirmed on dndbeyond.com under Basic Rules. For Clerics:

Ability Score Improvement

When you reach 4th level, and again at 8th, 12th, 16th, and 19th level, you can increase one ability score of your choice by 2, or you can increase two ability scores of your choice by 1. As normal, you can’t increase an ability score above 20 using this feature.

In general, ASI can depend upon your class as mentioned here:

When your character gains a level, his or her class often grants additional features, as detailed in the class description. Some of these features allow you to increase your ability scores, either increasing two scores by 1 each or increasing one score by 2. You can’t increase an ability score above 20. In addition, every character's proficiency bonus increases at certain levels.

For instance, Fighters gain an additional ASI at levels 6, 14, and 19.

As Willem correctly points out, multiclassing can effectively delay when a character receives an ASI.