I have boys that are 19 months apart. (now 6 months & 25 months) I used a Moby wrap (a friend made for me) and pushed the toddler in a single stroller or shopping cart. I would also recommend having an over the shoulder sling (like a Hotsling or Maya wrap) if you are in a hurry.
My youngest was born at the end of April. He would get hot in the sling sometimes. You might want to keep that in mind to help you decide on what you want to get.
I've also put the baby's carseat in the main part of the shopping cart & the toddler in the child seat for stores like Target or Walmart. I don't have a double stroller yet, but I was thinking of getting one now the younger one is starting to sit up. I didn't want to get one designed mainly around a newborn & infant carseat, since I don't go out that much by myself. I have 2 strollers & usually have someone else with me if we go to a mall or park.
It's a lot of work to take them both by myself, so I try to wait until my husband can go too. I expected it to be a lot harder having 2 under 2, but it wasn't too bad. The first 3 months were the toughest, but they went by so fast. Try to rest when they both nap. That was harder with having 2. Now that they are getting a little older, they are starting to play together & we're having more fun. It seems more daunting right now, but your oldest will be at a different stage when the baby is born. My older child did not have jealousy until after I went back to work. Then he started pinching, throwing toys, etc since he wasn't getting as much of my attention. At first he was kissing my belly & the baby. Most of the time he has been sweet & sometimes too lovey to the little one.
Another bit of advise, be careful where you leave the baby with the toddler around if you have to set the baby down to go to the bathroom or something. I always take the toddler with me or make sure the baby is safely out of reach. The toddler will sometimes try to throw or drop toys on the baby if I put the baby in the pack n play. The older child might try to pick up the baby or be too lovey & hurt the baby if he's in a bouncy seat, swing or on the floor.
Enjoy your time when they are little. This time it seems the youngest is growing up even faster than my first did.