I flew with jet2.com (Leeds) last year and took buggy and infant car seat. the car seat went straight to the over sized baggage area to be checked in before you went to security and the buggy went with us to the gate and even the steps of the plane. Our car seat fitted into a sleeping bag but you can buy specific car seat bags on ebay or amazon. If not, bubble wrap it which can be tricky if you are using it immediately prior to checking in!!! Also do not be put off with 'bulkier' buggies! I know jet2.com allow buggies to be separated into 2 parts - something I didn't know until checking in in which case I would have take my silver cross! On the same flight were a family with a Quinny and another with an i-candy. The buggies were also well looked after so don't listen to the very few negative stories. You will be glad to get rid of the car seat at check-in - believe me! Advice: Take a sling or baby carrier for the plane as there is no guarantee that you will see your buggy immdediately on disembarkation. I had a 45 minute wait at the carousel!!
You will need to check the car seat in at the desk but usually you will then have to take the seat yourself to the "special baggage" belt rather than it going on the conveyor with your cases.
I assume that you have paid for taking the car seat with you? Ryanair will carry 1 item of baby equipment for free, but if you are taking a buggy then the car seat is chargeable - do this before you leave because paying for it at the airport will cost a lot more.
What stage car seat is it? If rear facing, do you have a travel system you can use as your "pushchair" (ie car seat on frame) then just collapse at the bottom if stairs to airplane. If stage 1 I'm pretty sure you have to prebook this as a piece of luggage with Ryanair, although it has been a few years since I've done this) they aren't known for giving you anything for nothing!
I've never stored my car seat in a special bag, although I do have a spare one I use for flying which I'm not so precious about. Tighten up all the straps so they don't get caught on anything though.
We just flew with Ryanair this week and my car seat (maxi cosi) was attached to the buggy so we just collapsed it into its two parts and handed it over at the gate. It was totally fine at both ends of the journey, we didn't have it in any kind of protective case