To give you a summary:
As per operation - induction type or the autotype (either step down or step up)
As per function: Isolation transformer (1:1 turns ratio); step down or step up
We also have power transformers, current transformers (CT) and potential transformers (PT) used in electrical instrumentation.
for EE's we have primary distribution transformers and secondary distribution transformers (all induction type)
In electronics, we have the intermediate frequency transformers (IF) and RF transformers aside from the other types like
gate drive transformers, current transformers, isolation transformers etc.....