You have to start with organic chemistry and genetics. If you don't, you won't have a good understanding of what's going on.
Bioinformatics itself is huge, there are many many areas - protein folding, protein interaction, DNA/RNA interaction, all kinds of taxonomies, evolution, gene mapping, systems biology, etc etc etc.
I strongly suggest you pick something particular, because, otherwise, it's easy to waste years on learning many topics and not have any deep progress on anything in particular.
Bioinformatics itself is huge, there are many many areas - protein folding, protein interaction, DNA/RNA interaction, all kinds of taxonomies, evolution, gene mapping, systems biology, etc etc etc.
I strongly suggest you pick something particular, because, otherwise, it's easy to waste years on learning many topics and not have any deep progress on anything in particular.