+1 on trailers. Absolutely no doubt in my mind. I had a "kid Killer" seat for my first child, and it didn't take long for me to realize how dangerous they are. Hard to get the child in and out, the center of gravity on the bike is raised, they're so exposed. In short I hated it and she hated it. So I bought a used burley trailer from a friend, and that was an like an epiphany.
She loved it, and when we had our second child it was easy to tote them everywhere. The infant can ride in a car seat in the trailer until big enough to sit up on their own. when he was older I would take him to day care in the trailer, and lock it up there, and it had a rain cover. They really are easy to un-hook and hook-up. I would argue easier than wrestling them in and out of those kid killer seats. They trailer tracks behind the bike perfectly and they roll so easily. Never once did I even come close to tipping it over, and the kids enjoyed riding in it. Ours had the fold up front wheel and handle attachments to make it into a great stroller, I highly recommended that option. We used the heck out of that thing.
They are expensive for sure, but what price can you put on your kids safety and enjoyment? They will certainly outlast your children's need for it, and they hold their value so re-selling them is easy. Our trailer has been passed along to some friends and is now on it's third family and is still going strong.
Seriously...get a trailer. They are indispensible if you are a committed rider with small children. If that describes you, which I think it might, then I'm sure you won't regret it.
Congratulations on the birth of your second child and good luck!