Here are some tips, hopefully people will add their own!
You should never carry a carseat by the harness straps. However, you CAN use the top tether strap if your seat has one, it can be used to make a shoulder strap, and to attach it to a cart. But remember, just the tether, not the harness straps.
I get my carseat out of the car before flying and just play around with it. They sell my car seat at Target so I have also gone there and tried the seat on different luggage carts they sell, so see if I could secure it to the cart easily. Depending on the car seat, bungy cords sometimes work. (Early in the morning when there were few people around to think I was crazy.)
Most infant seats can be used with or without the base. I had one that HAD to have a base (bummer) but if yours installs without the base, play around with that before you go and perhaps you can take just the seat, no base. Some of them take practice to get a solid installation without the base so if you are not familiar with this type of installation make sure you practice before you go.
I bought a bag with handles for my carseat, but found that the bag was too bulky, it was easier to carry around without the bag. Mine has a handy bar on the back to use as a "handle". Depending on the carseat, I've found either attaching it to a wheeled cart, or just carrying it, has worked the best. But again, it will depend on your particular seat.