Agree that the thought of managing a Toddler and a Baby is daunting,but the reality is easier.Just need careful planning to save yourself time.
My advice would be to use the Buggy to transport baby,and all your essential bits to the Car,with Toddler dressed in Raincoat and wellies,which they love.perhaps get a wriststrap for toddler,which you could put over the handle of buggy,so your hands are free to push.
Get used to storing buggy in House somewhere.
I used to put TV/DVD on for Toddler after breakfast,whilst I took baby upstairs in bouncychair,and got dressed.
Alternatively,I now get out of bed,and get myself totally ready,before I wake up my DD's, (or undo the stairgate on their room),when I'm in their room I collect their clothes,and bring them and the clothes down.
I found it easier having a spare toothbrush and flannel downstairs,in the downstairs loo,if you have one,or available in the kitchen ,if not.
Lastly,I always packed my Changebag last thing at night,and get the bags ready for school/playgroup all ready that evening too,for the next day to just grab and go.
I thinks slings etc are good,but found it easier to load the buggy up.
It definitely gets easier.