I hope they do a total reboot of any characters, but not the basic background or setting. So the Federation, Klingons, and others are there and the basic concepts are there, but there's no Kirk or Worf or Janeway anywhere.
Also, and I know this will never happen, but I'd like to see a show more about the service of Starfleet rather than a particular set of characters or a particular ship. Starfleet has many ships, I would enjoy seeing it start out with what would have been an Oberth class ship, called something else in the new show with a new design, exploring new worlds, but not really having to stay with them. Maybe they run into trouble and get destroyed in episode 3 and the main ship of the series picks up from there, but then we switch back to other situations. I hate when they have every other ship be a redshirt ship, but I don't want a story where they are afraid of killing someone/thing because space is dangerous. And I'd really like to explore more of the other races that weren't, like Gorn, Tholians, Tellerites and many others, as well as meeting new ones.
I'd also like a lot more integrated Starfleet from the beginning and have more of the other peoples beside the Humans. Medusan navigator would be neat.
And another thing, I'd like individual episodes, but I'd like something that happens in an episode to not be totally forgotten, especially when it's a real setting altering event. For Example: This Side of Paradise, a cure all for just about every disease is found and never used or mentioned again especially when it was the central theme of the plot of a later episode, like Omega Glory or even Insurrection. I, Mudd there is a planet of fairly advanced androids found and they are never mentioned again, especially when Data is curious about being the only Android around. They tried one time to do something with the warp 5 speed limit but it was lame and forgotten really fast.
So, continuity with previous shows, not that important but nice, continuity with itself, very important.