To me, the important things in an election system are as follows (note that some of these overlap, and might be considered to be saying the same thing but in a different way):

1. Reduce the conflict between strategy and sincerity.  If voting in the way that is most in your interest feels dishonest, that is a bad thing. Although all voting systems can be "gamed" to some extent, some of them make it extremely hard to get any significant advantage by doing so. Plurality (our current system) is particularly bad in this regard.
2. Reduce the strategic value of parties.  Plurality forces people into opposing parties. Parties give their candidates an advantage, by making sure only a one candidate from their party is on the ballot, so they don't split the vote. While there is a short term, individual strategic benefit to those that join parties, the long term effect is polarized government, and a polarized constituency. Other systems are much more immune to vote splitting, so they won't have this effect.
3. Increase the tendency to elect centrist candidates. Elected officials that are centrist...that is, those nearer to the center of the ideological bell curve...have the best chance of serving the will of the people, and of cooperating well with other elected officials to actually get things done (especially if those candidates are centrist as well). Another way of saying this is that the system should tend to elect the candidate that is the first choice of the hypothetical median voter.
4. Reduce surprises on election day. The potential for surprises makes sports exciting, but elections and government are too important for this. We don't want a government that jerks us back and forth between right and left, we want one that drives right down the middle of the road with only small and smooth corrections. The only beneficiary of our surprise-prone system is the media, which profits from playing up the excitement of elections.
5. Reduce subjectivity. This one might seem odd, since selecting a candidate is by nature subjective. But given your preferences, what shouldn't be subjective is how you are to express them via your vote. In Plurality, it is subjective as to whether your vote means that you have selected your favorite candidate, or the one that has the best chance of defeating a candidate you prefer less. Other systems, like Approval, Majority Judgement or Range voting, incur subjectivity in the interpretation of terms like "approve", "reject," "good," "fair," and "poor," or in the numbers applied in the case of Range voting. Are they to be interpreted relative to the candidates on the ballot, or relative to some (possibly unrealistic) ideal the voter has in mind?
6. Reduce the need for voters to have knowledge of how others are likely to vote, or of the intricacies of the voting system itself. To vote most effectively under Plurality as well as Approval, you must have an idea of who the front runners are likely to be. Approval especially takes a special knack for understanding the most strategic approach to voting.
7. Have it possible to show election results in a comprehensible way. People are used to seeing the results of Plurality votes presented as percentages, a bar chart or pie chart. This is harder for some other systems, with IRV probably being the worst. Condorcet systems are challenging to distill into such numbers, but reasonable numbers actually can be produced.
8. Reduce tendency to flatten preferences into a single dimension. Under plurality, it is typical that the race will come down to two candidates, and a single vote can only move things toward the right, toward the left, or be effectively thrown away. This doesn't account for those who are social liberals and fiscal conservatives, who are libertarian, "green," pro-life and anti-war, or any of the other variations that just don't distill down to a simple position on the left to right spectrum. Better systems would allow a voter to actually express these preferences with their vote, while not throwing their vote away on a candidate that can't win.
9. Keep it as simple as possible for the voter. Voters should not have to learn how to rank candidates, approve a subset of candidates that surpass some arbitrary threshold, or anything else.  The ideal system would allow voters to simply pick the one they like the most, independent of who is most likely to win.