You usually check the car seat in at oversized luggage, they’ll give you a ticket for it and probably best to put in a bag if you can… could throw some nappies in there for extra padding… Pram take which seat is better for little one, they are usually wanting to look out more at 3 months so as long as you don’t need carry out for sleep and seat lays flat, probably take the seat… just tell them it’s 2 pieces when you checkin so both bits get a tag
Car seat bag, take to check in gate, they’ll tag it then you take it to the oversize baggage section (rarely will go with normal luggage at the desk). If buggy is handy to break down you can take to the gate and they’ll take it off you and put it under. Otherwise I’d get yourself a good buggy travel bag (does your brand make them?) and take to desk/oversize baggage too. Remember it’s only 2 pieces of oversize luggage though. Also use the car seat bag to stuff it with nappies (don’t take the piss I’ve heard some airlines are clamping down on this but saves a tonne of space in your suitcase!). This is the car seat bag we got, doesn’t do much in terms of protecting but keeps it all together and safe from the elements: https://amzn.eu/d/69FbWIf (shop around though!)
If you can afford it, you could try a Doona. It works as a car seat and stroller in one and can be used in the car without an isofix base. You can also get a bag for it. Then you can either put it in the hold at the gate (check with your airline) or use a baby carrier to get through the airport and pick it up at the other end.
Could you borrow/hire a car seat where you are going? There seem to be a few companies offering this service now. And what about a sling/carrier to move through the airport hands-free? That way you can check your pram in at the desk and deposit at outsize luggage before going through security. Have a great trip 🙂
Taking a pram and a car seat seems extreme. As others have sugggested - I’d be looking at options to make life easier e.g a doona, hire car seat etc. travelling is stressful enough without making it harder
We travelled for forst time in December with an 9 month old. We checked the seat in just after baggage and kept the pram up until the gate. It was all much easier than I'd imagined so try not to stress about it. Enjoy your holiday 🙂