DD1 was in daycare a couple days a week during my ML so I was going in there with a newborn and 23 month old. I carried DD2 in the infant car seat and DD1 walked beside me. We do the same thing now at 5 months and 28 months.
I didn't find it hard at all. I was able to hold DD1s hand with my free hand. Now she doesn't like to hold hands that much but stays right beside me. If you can't carry everything I'd probably use a snap-n-go with the baby.
I didn't start my kids in DC until DD was 2yrs and DS was 8 months, so he was no longer in the infant carseat. Initially I put DS in a stroller and had DD hold my hand, but I found it's quicker to just carry him. If I was you, I'd probably either carry LO in the carseat, or use a snap and go.
DS1 goes to a "mommys day out" once a week, and usually he walks beside me and I carry DS2 in the carrier. Sometimes DS1 decides he doesnt want to walk, and then I just carry him and the carrier in the other arm. It's amazing what you can do when you have to!
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#2 isn't here yet, but I won't plan on using a stroller. DD (19 mo's now) can walk just fine on her own (and Miss Independent is already too cool to hold my hand) so she'll walk & I'll carry the baby in the carseat. We don't have any busy parking lots of anything to worry about though - I can just park curbside against the sidewalk in front of the center, so no safety concerns there. I'll take DD to her classroom first, then drop the baby off which takes a bit longer (diaper change, form to fill out, bottles to put away, etc).
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Re: daycare dropoff and pickup
One of the moms at DD1's DC has her younger LO in a stroller at pickup and drop off.
Personally, I just carried DD2 in her infant car seat. And then later I would just sit her on the floor inside the DC if I needed to.