I am a delta frequent flier based in Boston. I actually find it pretty straight forward especially on Sunday's as long as its not vacation week from school in feb or April!
Even before TSA Pre-check, I routinely parked my car at Logan central parking 1 hr before my flights, got boarding passes, dropped luggage inside, cleared security, and traveled to the 2nd row of gates in time to get coffee at Starbucks and board my flight.
When traveling with the family instead of alone, we usually opt to be parking 2 hours before and entering the check in line 1:30 before...we are usually thru security with one hour to go and then proceed to the gate area where we get snacks, do final potty breaks, and the kids play in the play area while we pack up the stroller etc.
I recommend you check in online but print boarding passes as the kiosk and check luggage there...(I prefer that airport paper than mine). If you are coming in from the parking garage, when you come down the elevator it will be the section in the center of the Delta section. The Delta desk closest to the elevator is for silver status folks and the far end with more red than blue s the sky priority desk for those with gold status or above. When you are done with that, head towards security and follow the line for families if you can. They will coach your kids about their shoes (not taking them off etc) as well as have wider lanes for strollers and people.
For what it's worth, I dont Check in at the curb either....have. Good trip!!!