Forget the abstract concept of “nodes”. You’re using tiles, so just think of them like that.
You need a set of “open tiles” and “closed tiles”, and both kinds hold the position of said tile, the position of its parent tile, and the cost of getting to that tile.
When the algorithm starts, your open tile set contains the starting point. After that the algorithm will naturally expand outwards by querying neighboring tiles, which are very easy to find on a tilemap.