F.K.
I fly often with my two sons so I'm probably going to give you much more advice than you need but I'll try not to bombard you with too much info.
You can check the stroller before you board the plane. Be sure to ask for a gate check ticket at the gate before pre-boarding so you don't have to wait for it and miss pre-boarding. They don't tell you this, but you have to take the stroller down the jet way and right before you actually get on the plane, close or collapse the stroller and remove any detachable parts (cup holders or toys) as they can get lost or broken and the airline claims they aren't responsible.
You can bring an FAA approved car seat onto the plane for your toddler to sit on in his seat. The sticker on the car seat should say if it's FAA approved. I suggest you bring an infant sling/carrier such as Baby Bjorn or Wrap N Wear but the baby can not be in it during take off and landing. But it does come in handy so you can have your hands free while getting on and off the plane and to give your arms a break mid flight.
I also suggest you bring some disinfecting wipes on the plane and antibacterial wipes. You would be surprised at how much chewed gum I find on airplane arm rests or seats. Antibacterial hand gel is not allowed because it contains alcohol so airport security will take that away but the wipes are OK and the non alcohol gel should be OK although sometimes it's mistaken for the kind with alcohol and gets confiscated as well.
I also suggest you bring snacks and toys for you children. I often bring lollipops for my toddler (anything that encourages him to swallow often) and two bottles for my baby (one for take off and one for landing). I recommend you bring three toys per child, two favorites and one new one. Give her one toy at a time and switch them when your toddler gets board. Color wonder markers and coloring book come in handy except sometimes the markers roll away but you can tape string on them and tie the string to the book.
Check with your airline about checked bag fees because sometimes the fees are so expensive, it might cost less to have your family just buy you a car seat for your baby. Some airlines (like JetBlue) don't charge for the stroller, car seat or diaper bag if you are traveling with an infant but other airlines charge for checked car seats but not gate checked strollers and others charge for it regardless if it's checked at the gate so look into it before you travel.
Your toddler might have to take her shoes off to go thru security so you might want to dress her in shoes that are easy to remove and be sure she is wearing socks. Since I travel with two children (without my husband since he works a lot), I have a specific pair of shoes my toddler wears because he can easily take them off and put them back on himself and I usually have my hands full with my own shoes and my infant and my bottles of mommy milk which have to be removed from the cooler bag and opened and examined by security. Shuffling two children, milk, snacks, diaper bag and stroller thru security with impatient people behind me who often cut in front of me is the hardest part of traveling with small children. I have learned the hard way to close the stroller and put it on the x ray belt before I take off my shoes (ouch, that kick release latch hurts!) Once you get thru security with as much dignity as you can muster, the rest will seem easy.
Sorry for the excessive amount of advice but most of it, I learned the hard way so I thought I would let you learn from my mistakes. Have a safe and enjoyable trip.