Hi Nikki
Just some advice ive been living in zilina for around 3 years, which is a much smaller town than Bratislava. You will have no problem shopping I dont recommend tesco so much they have had a bit of a bad reputation here for selling product out of date so just keep an eye on the dates of what you are buying. They do have an english section dont know how big their section is in bratislava, where I am its quite small.
I recommend shopping in Lidl they also do an english week if you want things like uk bacon, you will get english cheese there weekly if need be. I primarely shop in Kaufland which is a very good supermarket and Lidl. I moved here when my boy was 6 months old, he goes to a slovak nursery but he is bilingual because I speak english and he speaks slovak in nursery, its up to you how youdo it with a baby, where I am there is an international school but your children will learn slovak aswell. Dont know of any facebook groups there is a website for mums I will check with my wife what its called. But if your children will be attending a international school then you would probably meet parents who are english speaking.
If you need any advice please dont hesitate to ask and I will help all I can
Huw