Depends on what it's for. A lot of people claim that a dishonorable is an automatic felony, that depends on a lot of factors and where you decide to live. Of course murder would be unanimous for a felony if adjudged in a court martial with a dishonorable because of what it was for. However Bergdahl was convicted of Military Only Offenses, some states would consider the offense of what he was convicted of and care, some wouldn't.
Either way what happens next, unless he waives it, is the discharge is not executed. He still has to wait for possible clemency from the Convening Authority (most likely won't happen), and since the offense were committed before 2013, when the UCMJ changed, the Convening Authority has the authority to disapprove the punitive discharge. While he's waiting for that he'll use any leave that he has, and be paid, until 14-days later when the demotion takes place (or the Convening Authority acts before, which probably won't happen).
After his leave is exhausted, he'll be placed in appellate leave status which is a unpaid status. He won't work, but he'll still be eligible for Tricare. He's waiting for the Army Appellate Court (ACCA) to review his case since he was sentenced to a punitive discharge, which takes about a year. If they affirm his conviction he can appeal to the United States Court of Appeals for the Armed Forces (CAAF) to hear his case, and possibly overturn what the ACCA and Convening Authority approved. If the CAAF affirm his conviction he can appeal to the Supreme Court via a Write of Certiorari, however the Supreme Court has a history of not wanting to hear a lot of military trials since their decisions could possibly change the Manual for Court Martial/UCMJ, which are written by Congress and the President.
A lot the JAG's feel that this was the right decision, as do I. Trump's comments as a candidate were stupid, but not technically Unlawful Command Influence (UCI). However affirming and talking about it the day prior to court martial starting, as the President, was very ignorant. Regardless of how one feels, if a Military Judge has to go on the record and hear and make a decision on the President's comments, that's never a good thing. Of course Trump's comments weren't going to affect the Military Judge, however the fact he considered it during sentencing means it was a mitigating factor for sentencing. And we all know him deserting led to his captivity, however he was still captive for five year in terrible condition. That was also another mitigation factor in sentencing.
I think a punitive discharge, no confinement was the right call for a sentence. However Trump's comments are going to create a ton of appellate issues.