I am at the end of a mission at it gives me the choice of Play Again or Continue. If I continue, can I play again later without starting a whole new Campaign?
Yes you can. All campaign missions are re-playable individually so you can play them again on any of the difficulties (for the sake of achievements,other story choices and side quests).
Same as you could in Wings of Liberty using the mission archives,since this is something that depends on the basic game rather than expansion.
The standard archives are unlocked after the 4th mission("Harvest of Screams" or "Domination").
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The master archives, available from the Main Menu (not during campaign), contain all missions you have unlocked, plus ALL possibilities for both mutations and evolution strands. So your choice during the campaign is only limited to that specific playthrough. Once you beat the game, everything is unlocked in a way you could have possibly played the game.
For example, if you chose to go to Char first, you could not build roaches. But if you completed both Char and Kaldir, you could go to the Master Archives and play on Char with Roaches and a stronger Kerrigan. If you completed Zerus as well, Kerrigan wil be even stronger at level 35 with a choice from all three abilities of each tier.
So no worries. Even if you beat the game on Casual mode, you will have unlocked everything and can replay any mission in any possible configuration.