What are the best transition times in your age group? I mean, you did 3:21 and 3:02, but we don't know whether the fast guys did either <1:00 or >2:30

I normally have some of the fastest transition times at my races, mostly sprints and olympics with some long course. So here are my thoughts:

Minimalism - your transition space should look sparse compared to those around you - especially in sprints and olympics you really don't need a lot of "stuff".
T1 - Shoes on the pedals - you don't need to do a flying mount, see the thread with videos of different bike starts. So the only thing you have to do is put your helmet on, grab your bike and run - so much time saved not touching your shoes in any way.
T2 - practice pulling feet out from shoes before dismount line and leaving shoes on pedals. Again you don't need to do a full running dismount, you can come to a full stop, climb off the bike but then you are running with just your bike to handle. Rack bike, helmet off, shoes on, grab bib belt and start running before you put the belt on. You don't need anything you can't pick up and run with.

Bonus Points:
Be running everywhere - the top place people will be.
Socks - if you wear them make sure you have picked a pair that you can get on easily, and practice with wet feet and hands.
Mounts/Dismounts - these don't change your T1/T2 times but there is free time there for your bike if you practice these

I normally have some of the fastest transition times at my races, mostly sprints and olympics with some long course. So here are my thoughts:

Minimalism - your transition space should look sparse compared to those around you - especially in sprints and olympics you really don't need a lot of "stuff".
T1 - Shoes on the pedals - you don't need to do a flying mount, see the thread with videos of different bike starts. So the only thing you have to do is put your helmet on, grab your bike and run - so much time saved not touching your shoes in any way.
T2 - practice pulling feet out from shoes before dismount line and leaving shoes on pedals. Again you don't need to do a full running dismount, you can come to a full stop, climb off the bike but then you are running with just your bike to handle. Rack bike, helmet off, shoes on, grab bib belt and start running before you put the belt on. You don't need anything you can't pick up and run with.

Bonus Points:
Be running everywhere - the top place people will be.
Socks - if you wear them make sure you have picked a pair that you can get on easily, and practice with wet feet and hands.
Mounts/Dismounts - these don't change your T1/T2 times but there is free time there for your bike if you practice these