Starcraft – Will I be able to see the results of the previous campaign if I start a new one

The campaign lists 26 total missions, though some of these are "A or B" missions. Though you may only complete one of these missions for credits / research / campaign credit, the other, alternate mission is available on the console on the bridge to be replayed any time you like for achievements, etc.

Given the nature of those "pick one" missions, I believe there are 29 total possible missions +- a bonus mission that is only unlocked during special circumstances. (Or there may be 26 missions +- the bonus mission, I'm unsure how the "Campaign progress" bar, x/26, accounts for the branching scenarios.)

There is one mission like that in the final few missions that give you an option between 2 missions and depending on what you choose it makes the next mission change a bit, but it is still the same mission.

There is also one hidden objective in one of the missions that will give you access to a secret mission that can only be played by completing this objective, this mission however is not crucial to the story.