You understand the way to boost ability scores correctly
At the beginning, when you make a character, you apply the ability score boost due to race after you have assigned the numbers to each of the six abilities.
When Ability Score Improvement(ASI) comes at 4th level, 8th level, (and 6th level for Fighters) and so on, you can apply two points to any ability score at that time when you have gained 4, 8, (etc) levels in a class.
As an example, a human barbarian
On page 49 of the Players Handbook, the Barbarian class feature of Ability Score Improvement tells you that you can at levels 4, 8, 12, 16, and 19 increase one ability score by two points, or two scores by one, and none may be increased beyond 20. Each class has an entry for ASI, so the player needs to read the ASI entry for their class. (It's different for Fighter, for example.) Let's say that our Barbarian was a human who rolled the following raw scores for abilities.
Str 15 Dex 11 Con 14 Int 10 Wis 11 Cha 10
A human gets to add one to each ability score.(The racial boost you use at character creation).
Str 16 Dex 12 Con 15 Int 11 Wis 12 Cha 11
At 4th level, the barbarian gets to add two points(ASI).
Either add two to one score, or add one to two scores.
Examples of making that choice:
Str 18 Dex 12 Con 15 Int 11 Wis 12 Cha 11 (+2 to Str)
Str 16 Dex 12 Con 16 Int 11 Wis 12 Cha 12 (+1 each to Con and Cha)
If any of your players wish to multiclass, ASI is based on class level, not total character level.
For example: a third level Barbarian who chose a level in Fighter when enough experience points were earned for 4th level (2700 XP, p. 15, PHB, table) would not get an ASI, being Barbarian (3) Fighter (1). He would need to earn 4th level in either barbarian or fighter before getting an ASI.