Another way people can check is by disregarding whoever is involved in the 'and'.
You wouldn't say 'Johnnie saw I going to the shop' [unless you were in the West Country, maybe], so don't say 'Johnnie saw my brother and I going to the shop'. He saw my brother and me.
In the same way, you wouldn't say 'Me went to the shop' [unless you were about three years old, possibly], so don't say 'My brother and me went to the shop'.