If you go through basic training, what kind of conceal carry training do you need to still take?

All of them... Some courses will let you basically walk through if you have military experience but just because you have served doesn't mean any state is just going to hand you a CC permit.

1. Depends on the state you get your weapons carry license. In some states there are more stringent training requirements than others. Some states have no training requirement. Some states have Constitutional carry.
2. Regardless of any training requirement, it is always beneficial to get more training, advanced training, and practice.
My $0.02.

In states that require training, most of them still require you to take the class.