Not sure how old your kids are but if you have one remotely close to stroller age my advice is this. After you check-in at the Buffalo airport you need to go through security. Plop your kid(s) in their stroller and go to the far right of the security line. There is usually no line and a person will be seating and/or standing to look at your boarding pass. This is for guests requiring assistance ONLY. Families with small children in strollers, wheel chairs etc...
We had some fine gentlemen workers walk by as we were getting in the BIG line-up with everyone else when they told us to go to the right. It was a time-saver, so handy when you have small kids. We were traveling with another family so a few kids! We were so thankful and those guys didn't have to say a thing so we really appreciated it.
We've done this twice now since learning this tip. A young couple followed us on our last trip and tried to get through b/c they were going to miss their flight. The lady made them go back to the other line with the bazillion people in it. The regular line actually moves fairly quickly it just looks bad, but the ease of getting through quicker with kids is great.
Another tip! Just to save a few bucks (and who doesn't want to do that

)They have curb side check-in. DON'T use it. The luggage handlers require a tip. Why? I have no idea but let me tell you they really work it until you tip them. Just say you'll carry your own bags inside and check-in inside. It's usually not busy.
Are you leaving on an early morning flight? We usually stay at the Comfort Suites on Dick Rd the night before. Our flights are always early morning flights. They have a continental breakfast, shuttle service (literally leass than 5 min away), AND you can park your car there for up to 2 weeks (included in the price of your stay).
We've flown Delta once before and had no problem. Have fun. Buffalo is a great, small, clean bright airport!