Thanks to a misreading of Bernard Williams, many modern virtue ethicists say that Socrates posed the question “How should one live?”, but that's not quite true. The question that Williams actually refers to, as contained in my edition of The Republic, 352d, is “whether the just live better and happier lives than the unjust”. Coinage of memes by misinterpretation happens all the time; it's a valid process. And the question of how one should live is a valid one, but it's usually more helpful to break it down into pieces.
Richard Hamming pointed out, in You and Your Research, that if you want to be a great scientist, you have to work on great problems. This has been a big influence to me, but the bigger question that it only partially tries to answer is: what is the best way to contribute to the general lore of mankind? Are you more likely to contribute good things in the arts, the humanities, or the sciences? What about being interdisciplinary? Should you try to do all things and then choose whatever you turn out to be best at? Or is it worth cultivating skills in a particular area?
The Hamming approach is focussed only on the sciences. In Magic, Science, and Civilization, Jacob Bronowski mentions that one of the discoverers of DNA, one of the Nobel Prizewinners, wrote a book about how they specifically set to work on the most challenging problem of the time, so as to maximise the reward; that they literally focussed on winning the Nobel Prize. Another Nobel Prizewinner, Feynman, writes memorably that he wanted to reject his Nobel Prize because it was just nothing to do with the work. The work was its own reward. Presumably Crick et al. felt the same thing (I haven't read that book), but the two mindsets on how you focus yourself are interesting.
The interplay of value and recognition is very tricky. Newton worked on developing calculus before Leibniz did, but Leibniz was the first to publish, and now after some controversy as to who was first, they're jointly credited. But the integral sign, the dy/dx notation, and even the name “calculus” are all from Leibniz, even though Newton got there first. If nobody reads your work, it is functionally useless to humanity; but that doesn't prevent you from historically being seen as having the greater insight. And you might not be read not just because you refuse to publish, but because nobody listens to you when you do publish.
Only a handful of scientists make the really big discoveries. Are the great scientists really great, or just statistical freaks? Can amateurs contribute usefully to science? These are two questions, quite challenging to the Hamming ethos, that I've been asking myself a lot. It's not so easy to be a productive amateur in science, I think, though there are some people who do very well. The Rev. Robert Evans, for example, holds the record for the visual discovery of supernovæ, having found forty since he started in 1955. Bill Bryson wrote about him in A Short History of Nearly Everything, but for some reason when I found the Supernova Search Manual online a few weeks ago, I didn't connect immediately that it was written by the same guy.
But in general, how is an amateur to contribute usefully to science? As well as chasing supernovæ, you can do wildlife surveys for ornithology and botany, and perhaps discover a period 19 oscillator in Conway's Life if you work at it hard enough—things like that. But the humanities beckon strongly as being much richer grounds to mine for discoveries and general contributions, especially to the amateur.
To take myself as an example, my amateur contributions include having: independently found a link between Hamlet and Jabberwocky; contributed dozens of antedatings of words to the OED, including words like “meteorite” during the course of my essay on the discovery of what meteorites are; found a possible Goidelic and Brythonic hybridisation in Cumbric; and underlined the connection between pillow mounds and toponymy. Moreover, I've found out so many things about Shakespeare, who is generally considered to be an overstudied subject, that I'm preparing a large work of these findings. That even Shakespeare is not, in fact, overstudied shows just how much great work there is especially to be done in literary history.
An interesting side-effect of being a literary historian is that you get to study how great figures—from Dafydd ap Gwilym to Thomas Nashe through Sir Thomas Browne and Newton and Coleridge to G.K. Chesterton—came to be regarded as great, and to form your own opinions about how great they were. And, of course, to perceive whether there is anything to be learned from them in answering the question of how best to contribute to the pile of humanity's lore.
One of the greatest principles that is springing up again and again in my own studies is the reinforcement of the idea that big advances are indeed just statistical freaks, and that it's best to instead pursue what I call the Wainwright approach of making small atomic contributions in great number. The burden of trying to find one big insight leads to a kind of intellectual anxiety, and even if you're fortunate enough to have an insight, you can end up then trying to replicate it in the same area rather than continuing to make other good advances (the Semantic Web may be a good contemporary example of this syndrome).
With Coleridge especially, though he's widely regarded for his mystery poems of Kubla Khan, The Ancient Mariner, and Christabel, I think that the power of Coleridge is in almost everything he did, from the grandest schemes to the most petty minutiæ. Part of what it means to be a good Coleridge scholar is sorting out all the stuff worth propagating from his tremendous output, and synthesising the whole in terms of the original notions, received notions, and augmented notions of further comment. We need, in my opinion, more people coming to read and understand his Letters and Notebooks, his great prose works; that is my vote as an historian (cf. “The Very Consideration” in Five Essays).
As well as science and the humanities, there's also art to consider. The value ethics of art are particularly complicated, and I don't want to get too much into art philosophy here. But I certainly think it's worth trying to cultivate artistic tendencies alongside other things, because as individuals one of the most interesting things that we have to add to the pile of lore is the unique aspects and perspectives that we have, of which art is a more positive expression than what we can contribute through the humanities and science.