What sort of surgery will I need to get in order to correct a negative canthal tilt and obtain almond eyes? (photos)

Changing eye shape is complex. Not sure if you need surgery but best to have personal evaluation to determine what is bothering you and what can be improved. 

Hi 
When we talk about canthal position we talk about the position of the lateral canthus in relation to the medial canthus. A "negative tilt" generally refers to the lateral canthus being lower than the medial canthus. In general most individuals have a lateral canthus slightly (few mm) higher than the medial canthus. This does not appear to apply to you based on your photos. 

All the best!