Either way, you'll want to create a companion JS file with a function in it called PreSaveAction().
Run whatever validation you want to run in this function. Basically, if everything is validated OK you should return "true" and if there's anything wrong you should return "false".
Unfortunately that is the best way that I know of to do this. There isn't a switch you can flip in Settings or wherever that says "validate everything all at once, not just Picker controls". Also, just because you're doing client-side validation doesn't mean that SP won't still do the same server-side stuff it's currently doing, so if you change a business rule in your JS function but don't update it in SharePoint you will probably have validation issues.