Totally understand your anxiety about the responsibility. I am lucky that my DH is almost always with me on grandchildren 'duty' day and he does the driving. I have steered (sorry!) clear of driving with the GC in the car and feel his bigger car offers more protection as well. I notice you say you have a travel cot - if you find this difficult to reach down into you might want to invest in a proper cot or perhaps toddler cot bed that will grow with the child? I have bought everything (apart from car seats) off pre-loved sites (buying new mattress), eg cot, high chair, buggy. I found it best to have our own buggy/stroller to avoid finding ourselves without one when parents drove off to work with theirs in the boot.
We visit local parks and soft play areas - always somewhere where we can have a coffee too. Now that oldest GC has started school and we do the school run once a week, I find that even more nerve-wracking as he uses his scooter and is some way ahead of us. He has been well trained to stop at every kerb but even so, I am a nervous wreck and have abdicated this duty to DH most times too while I stay with the younger one. I have also found the small backpack with rein attached a brilliant idea as children always very happy to have it on 'like Daddy'. Enjoy - it is over all too soon but hopefully lays the foundation for a long-lasting bond.