Say Mark is charged with A (let's say Murder), but the charges don't stick because e.g. he didn't do it on purpose. Can Mark then charged as a follow-up for B (e.g. involuntary manslaughter)? Or does getting exonerated from A mean you can't keep charging him with lesser crimes like B until something sticks? And if you can charge him with B, does that mean you have to do a whole nother trial?
It depends on what you mean by "doesn't stick." If Mark is acquitted of murder in state court, he can't be tried again in state court for the same act of homicide. If the charges are dropped or dismissed without prejudice before jeopardy "attaches" then Mark can be charged again and tried for manslaughter or even murder.
The Double Jeopardy clause encompasses four distinct prohibitions: subsequent prosecution after acquittal, subsequent prosecution after conviction, subsequent prosecution after certain mistrials, and multiple punishment in the same indictment. Jeopardy "attaches" when the jury is impanelled, the first witness is sworn, or a plea is accepted.
Any lesser crimes that could be charged for the same act (or acts) are instead considered concurrently with the main charge (or charges) at the original trial. See lesser included offense at Wikipedia.
And if you can charge him with B, does that mean you have to do a whole nother trial?
Yes. If you can charge Mark again, this means that jeopardy did not attach. In this case, if a new charge is to be brought, the entire process starts again from the beginning.
Criminal Code, s. 662 provides that the accused is at jeopardy of all lesser included offences that are subsumed within the offence charged or as described in the count.
An offence is “included” if its elements are embraced in the offence charged (as described in the enactment creating it or as charged in the count) or if it is expressly stated to be an included offence in the Criminal Code itself.
(R. v. G.R., 2005 SCC 45, paragraph 25)
Where the accused was at jeopardy of a conviction on a lesser included offence, the principle against double jeopardy (specifically, autrefois acquit) precludes a subsequent prosecution on that lesser included offence.
The CPS offers guidance on this. You can't be retried for a lesser or more serious offence unless "new and compelling" evidence comes to light.
Under normal circumstances they get one bite of the cherry.
Retrial of Serious Offences
If a person was acquitted of a qualifying offence within section 75 (1) [of the Criminal Justice Act (2003)] then that person is treated as also having been acquitted of any alternative offence which is itself a qualifying offence, i.e. an offence of which he or she could have been convicted in those proceedings because of the offence being charged in the indictment...
The defendant cannot be put on trial twice for the same crime*. But the judge can derubricare the crime, which is a technical term to say that the judge can reduce the charge.
[*] After the first degree the prosecutor could challenge the verdict and trigger a second degree trial, but that would be something different.