Dasyam
Dasyam means regarding oneself as a servant of the Lord. This attitude is the source of actual freedom and unlimited bliss. How is that?
The soul is by nature tiny and needs to take shelter. In the material world the soul takes shelter of material nature and consequently becomes conditioned by it. When serving the Lord directly the soul takes shelter of Him and thus becomes free from all material conditions.
Dasyam could mean menial service like in pada-sevanam, but it could also mean serving the mission of the Lord. The mission of the Lord is to take all conditioned souls back to the spiritual world, the eternal abode of the Supreme.
Sakhyam
Sakhyam and atma-nivedanam are advanced methods of devotional service. Sakhyam means regarding oneself as the Lord’s friend. It is a specific kind of relationship with the Lord, in which there is much less reverential attitude than in dasyam.
Sakhyam cannot be practiced in the beginning stages, but the tendency for it must develop on its own accord.
Atma-nivedanam
Atma-nivedanam means surrendering oneself (one’s atma) completely to the Supreme. It is not an easy thing and can only be practiced by special souls.