The direction of the field lines shows how a compass would line up if placed at that point. We can use the right hand grip rule to remember the relationship between current and field around a wire:
An alternative way of remembering this is called the corkscrew rule: when screwing in a corkscrew to a bottle of wine, the point indicates the direction of the current, the the turning the field direction.
N.B. This does not, of course, work with left-handed corkscrews!