I'm a relatively new player and am confused about ASI.

My DM is having us start at Level 10 and using standard array for the ability scores. I'm playing a fighter, and my question is: do I calculate the ASI fighter gets at level 4 nd 8 into my ability score, or ignore it completely?

You'll select those as 4th and 8th level features (separate from the initial ability score assignment) if you're building the character through D&D Beyond.

Create your character at level one and then input you stats, go all the way through until you see the character sheet.

After near the avatar and the character name there will be an arrow with a drop down list, click on "Manage levels".

There will be a "Level" drop down that you can increase, if you do so one at a time you'll see that it adds your new features and you can see any that require more input as they will be marked with a blue/white exclamation mark.

I agree with the other posters....make your character here on dndbeyond one level at a time. It will help you to make sure you don't forget something.

First, choose your stats, allocating from the standard array.

Then add racial modifiers

When you get to levels that grant ASIs, add in those bonuses.