There's not one right answer. Different prepositions can be used for different meanings. "Towards" is probably the most neutral choice here, as it carries the least extra connotations.
The infuriated bird suddenly doubled back on its course and charged
straight after me.
"After" implies that you are actively moving away from the bird and it's trying to catch you. The image this conjures is that you are running from the bird and being chased.
The infuriated bird suddenly doubled back on its course and charged
straight at me.
"At" implies the bird ran in your direction and you were its target.
The infuriated bird suddenly doubled back on its course and charged
straight towards me.
"Towards" implies that the bird ran in your direction, whether or not you were its target (maybe it was charging at something close to you instead)
The infuriated bird suddenly doubled back on its course and charged
straight into me.
When "into" is used with a single person, it carries the meaning that the bird actually collided with you, rather than merely approached you. (With a group, "into" would mean that the bird made it inside / among the group of people).
The infuriated bird suddenly doubled back on its course and charged
straight to / up to me.
This doesn't work very well with "charged", but it implies the bird ran towards you and then stopped next to you. It would be more common in a sentence like "My dog ran up to me and begged for playtime".