How early in the immortality cycle are people? It would probably take several decades, if not centuries, for people to truly understand what immortality means and that they actually have it (i.e. if I gave you a shot today and said it made you immortal and unageing, how could I prove it and how long until you actually believed it?). So for mortal-now immortal people they would still retain an understanding of mortality, how fleeting life can be, and the pressure of time. But for immortals who have never known mortality, they may be very casual about time, yet also extremely risk-adverse, since they can expect to live forever unless they have an accident. How well do they heal? I think a lot of our behavior WRT sports, drug use, dietary habits, etc is predicated on the notion that we all die eventually so "life is for the living". But this would no longer be true if we had immortality. We also have a lot of genetic pressure to reproduce, how well could we reconcile immortality and family?
Many sci-fi works deal with functional immortality (transfer of consciousness into a clone body, deaging procedure, etc). Some tidbits from them include fixed time periods for marriage (since 'til death do you part' is a REALLY long time), potential wage slave status to pay for life prolonging treatments with class stratification based on those lines, and cultural stagnation as immortals presumably cap out on the types of new experiences they want and instead dwell in a persistent form of nostalgia. Some concepts for REALLY old folks include selective memory dumps in order to be able to function with off-site storage of memories or notes for the immortal to consult (the room of diaries for the immortal girl in Dr. Who, for example).
The old folks I know often seem to reach a period of resignation and acceptance of death. But how much of this is due to an infirm body and loss of companions versus some biological clock winding down, who can say. There is certainly a component of "old man yells at clouds" WRT older generations and new technologies and culture that may or may not persist in immortal society that may also hasten a desire to end life (this is frequently seen when there are a few immortals amongst lots of mortals). It certainly will be critical to have new frontiers to explore due to population pressures (and long term interstellar travel wouldn't be [as much of] an issue, reference "The Boat of a Million Years" by Poul Anderson).
Eventually you'd like to think that immortals would all become like Bill Murray from "Groundhog Day", accomplished in many cultural arts due to the large amounts of leisure time they would have to master skills. A better appreciation for the delicate balance of life on Earth would be nice as well, since you would be around to see the loss of biodiversity and climate change.