just learned need head gasket - oil leaking into radiator -- how serious is this

Was told oil getting into antifreeze -- was shown oil on several hoses

Transmission fluid IS oil... It all looks the same when mixed with antifreeze! .. I'm just stating what the most common cause for this is... Transmission fluid is constantly circulating 'under pressure' through the fluid cooler inside the radiator when the engine is running, so a leak in the cooler is simply the most likely culprit... That's what I have found over the many years of auto repair I have been through.. Good luck to you, whatever the outcome.