My personal view is that immortality is something everybody should aim for -- unless immortality is invented before I die, I plan to get myself cryonically frozen and revived in the far future.
But in any case, here are a few of my thoughts on immortality, things to consider:
"Death": In real life, many people believe that death is something irreversable that alters your body -- it's more likely that human life is the gestalt realization of all your neurons firing together, and if those neurons turn off, turning them on again won't change a thing, since they'll have remained in the same physical state. It's like how computers remain what they are even if they lose all power.
Of course, this assumes that there's no soul that is lost upon death. I don't believe they exist in real life, but they can exist in your conworld. Some interesting ideas can be derived from this -- for example, a machine that you use to collect the soul of a person when they die, so you can put it back in after their body is fixed.
Nanotechnology: Immortality through nanotech would be done through repairing cellular (particularly neurological) damage. This is the most likely way that immortality will be done in real life.
But having nanotechnology that is strong enough to repair bodies implies many things -- for example, if they can repair bodies, they almost certainly can change them too. Transhumanism, in other words. Most diseases and injuries could be readily cured, as well.
The true capability of nanotechnology as a whole is much debated, so be careful how powerful you make it -- you could end up making your conworld unimaginably powerful and unfamiliar.
AI: AI is almost certainly required for technological immortality, since it would be very difficult to do with solely human minds. Again, the true capability of AI is controversial, so be careful.
Ego and Morph: A concept commonly used in transhumanist fiction, the basic idea is that if your body dies, your mind can be reuploaded back into another body from a regular backup or a rolling backup located in your body.
This requires several technologies: advanced neurology, genetic engineering, detailed psychology, and uploadable intelligence and memory.
The most interesting implication of this is that your mind is a separate thing from your body -- your "ego" and "morph", respectively. Expect this to have shittons of social effects.
Getting a good quality morph would be important, as well, so if your conworld has social stratification, expect lots of discrimination in this regard. There will likely be morphs designed for various situations and with different capabilites. (A morph for zero-gravity life, a morph to survive on the partially terraformed Martian surface, a morph for being attractive and good at sex...)
There's no reason to restrict yourself to human forms, either. Animal morphs are going to be common, and completely original forms will be popular among eccentrics.
Other implications include that if your memories are recordable, they should be replayable. Bam, you've eliminated the concept of forgetting, and sharing memories and experiences would suddenly become a new form of media. You might even be able to edit memories, which would be pretty interesting -- especially if you're original and don't make it into yet another dystopia.
Finally, this would make psychological damage that much more terrifying -- if your body is damaged, you can get a new one. But if your mind is damaged...
Synthmorphs: There's no reason your body has to be purely biological -- if it's possible to digitize minds, then having a robot body -- or, a "synthmorph" - would become plausible.
This would make you immortal by the virtue of the fact that your body is insanely tough, and would let you survive a lot of things that would be fatal for a biomorph. If synthmorphs are cheap and biomorphs expensive, there will be a prejudice against synthmorphs, being associated with poorer classes who have to work.
Again, an even greater freedom for non-human forms.
Infomorphs: Why even need a body? An infomorph is an intelligent mind that exists purely in virtual space, having no body. Cool, innit?
If infomorphs are invented early, expect them to dominate vast areas of technology and work, since they could control machines and use data without having bulky physical interfaces.
Some AIs could be considered effective infomorphs.
Magic: I can't really give concrete advice, here, since magic systems vary so broadly, but remember to consider the implications and side-effects your immortality magic has. Of course, the same applies to all magic.
Language: The idea that language will stagnate if everybody's immortal strikes me as unlikely. The main reason old people don't learn youngster slang is because of mental degradation in innovation and learning ability, which would be fixable with nanotech, so the perpetually young immortals are going to be continuously inventing new words and phrases as culture and technology moves on. Consider the internet that exists now, where slang and memes from the 00s are seen as quaint and boring by 10s interneters, even though most of them are the same people who used those memes.
Immortals might also deliberately change language. Tolkien envisioned the Elves as perpetual language tinkerers, who perpetually changed the language to be more interesting and aesthetic. More science fictionally, the ability to edit thoughts and memories would lead to being able to learn a language very rapidly, if not being able to simply upload it into your brain. This would lead to exended language diversity -- why learn a standard language if it's so easy to learn the other language you're dealing with now?
Language editing will lead to subcultures inventing their own local cultural language, a sorta hybrid of cant and jargon. Conlanging would become a profession, since a lot of groups, from clique-y punks to army units, would want their own personal language.
Boredom: I don'þ know if immortals would really be bored with life, but I lean towards "no" -- the universe is so vast, and human culture is so diverse, that I don't think I'd be bored with it, ever.
If that doesn't convince you, consider that the "bored immortal" is one of the most common cliches used. Wouldn't it be a fresh twist to have it different?
Birth rate: It's well-attested that the more prosperous your country gets, the less children they have. Germany and Japan are having serious problems with this in real life.
So, an immortal society would be very slow growing -- people would have children once or twice every few hundred years. This would also make children a very small minority, in contrast to our world.
So, yeah. That's what I could think of that's relevant to immortality.