Is It Better to Do Braces First and then Crowns or Vice Versa?

Great question and since I am doing braces for a third time before I do my dental work I can't emphasize how important it is to have your teeth in alignment and your bite correct before any care.  The only exception will be if you have decay,  then you want your dentist to do a filling but not any crowns until your teeth are in the correct position.  Hope that helps!

I think doing the orthodontic treatment (braces) first would usually be the best idea. Having said that, everyone is different, your dentist and or orthodontist are the ones who really should determine the order of treatment.
 

A prudent question might be; "What are the crowns for?".  If the crowns are for cosmetic reshaping of the teeth, then definitely wait until after the orthodontics.  You and your dentist may find that the crowns may not be needed or something less invasive such as cosmetic bonding or porcelain veneers may be more ideal.

Orthodontics can take years to complete.  If significant decay exists, the decay can advance during the orthodontic process and destroy the teeth. If significant decay exists or the existing dental restorations in your teeth are problematic or mis-shaped, then the dentist can/may create temporary crowns or composite tooth buildups to protect the teeth while the orthodontic treatment is being done.  The bottom line is that is rare to ever place a permanent restoration until after the braces are off.