Whilst your on holiday accept that any routine (food, sleep) will go out the window but that your dc will settle back in fine when they get home.
You should be able to take your buggy to the gate/plane door, and may get it back at the gate though it might be at luggage pickup (often on a different belt to your suitcases). A sling/baby carrier is a good idea for airport and for sitting on plane if 1 year old isnt too wriggly.
Re food for the plane, lots and lots of snacks - cherrios, breadsticks, raisons, anything they like or is a "treat". Just keep feeding them! (See above re routine....). If breastfeeding do it for take off and landing as the sucking helps their ears, alternatively a dummy works too. At 1 they probably will be off formula and on milk but I used to take a couple of cartons of the ready mixed stuff on the plane if it covered milk time, as it didnt need to be kept chilled like cows milk. If you need lunch/dinner cheese, yogurts are good. Or revert to an Ella pouch. If you have a couple of jars/pouches of baby food airline security will often ask you to pick one to try. If you need to keep stuff cold, dont use the freezer packs as I dont think they are allowed through security but use a small pack of frozen peas in the lunch boxas it will warm up slowly.
Check with you hotel if they offer sterlizers, microwaves, most family places are pretty good at this.
Take a packet of weetabix/whatever they like for breakfast in your suitcase so you know they will have a good start to the day (and if they only eat fruit for the rest of the day then its less stressful!)
If you are going to be using a travel cot from the hotel take one of your own sheets so it smells like home.
And have a lovely time!