Title kinda says it all.
Assume I was convicted of a crime in (say) Illinois and also convicted of a different crime in federal court. Whose prison do I go to first? Is it simply whichever side gets sentencing done first?
Whichever side is the first to lock you up to serve the sentence.
You’d need the wisdom of Solomon to solve this one… 
Unca Cecil’s 1979 column:
I would imagine that Trump couldn’t be in trial for both things simultaneously since he has to be physically in court for trial and he can’t be in both places at once. So it would be whichever one puts him on trial first. If it were Georgia that went first, then he’d serve prison time for Georgia charges first.
I presume the clock on the second jurisdiction’s incarceration doesn’t start until the first jurisdiction’s term is satisfied. If the feds give you five and the state gives you five and goes first, the feds wait for their five until the state is done and hands you over.
In a recent example, the killers of Ahmaud Arbery were convicted and sentenced to life for both state and federal crimes. They attempted plea deals with the feds to arrange to serve their sentences in a federal prison. The judge rejected the deal, but it would appear that it’s at least possible to negotiate it.
Could set up an interesting constitutional issue, if Trump is convicted but manages to get a deal like this somehow (possibly with the help of a Trump appointed judge), can someone convicted of a state crime but serving their sentence in a federal prison be released because their federal crimes are pardoned? Could a re-elected Trump order himself freed from federal prison?
But if the defendant is released on bail rather than put into pre-trial detention, then it’s perfectly possible to attend two trials in parallel, as long as the actual hearings don’t occur on the same dates. It’s also possible for trial 2 to begin later but be concluded (both verdict and sentence) sooner than trial 1.
The length of the sentences may also have some bearing on which one takes precedence. In this case, the killers were sentenced to life with no possibility of parole, plus another 20 years. I believe the extra 20 years was on the federal charges. Since that sentence will never be served, it’s pretty much moot. They were just looking to make the additional federal charges an excuse to spend the first 20 years in a more comfortable federal prison, a request that the judge quite properly rejected.
It varies, depending on the state and the crimes committed. I had a fair number of my patients who finished their time in our state prison system, then were ‘released’ to a federal penitentiary to start serving that term, and vice versa.
I did note that generally, the more severe crime conviction often got served first. Which makes some sort of sense, start doing your time in a maximum security facility then move to medium/minimum facility later on in those cases.
If you get sentenced to “life and 20 years” do you serve the 20 years first?
It’s your choice. The prison system is very accommodating on this.