It my be useful if you copy the macros you are having trouble with in here and then people might be able to see if anything is set up wrong with them. I'll try and explain some basics, but I apologise if I'm repeating stuff that you already know.

As for which macro goes with which character, split it by character, anything you want your priest to do goes in their macros (personally I use the character specific macro section unless I run out, if you have any macros which you might want to use for a different 'team', e.g. the 'accept all' macro then put that in the general macros tab so you don't have to rewrite it for other characters on that account). For starting out it's probably best to have one toon as your 'main' (in this case your mage I think as that's how you seem to have set it up) and the other toon as your 'alt'. So your main is the one who's screen you focus on most of the time and the alt is the one who's in the background and will need most of the macros to get casting. Once you're more comfortable with macros and multi-boxing you can move onto a more flexible set-up.

For simple spells your mage shouldn't need any macros, as that's the one you'll be actively controlling when fighting stuff.

Your priest will need macros for their abilities to tell them to target what your mage is attacking. There are several ways of doing this through macros and I'll explain the most common ones here (I'd suggest you pick one way and use that for all your attacks, rather than mix and matching). Your alt can have a delay in picking up your mains target though, so when your priest doesn't attack it might just be that it's being slow to pick up the mages target.

Focus:
Very simple to use, you focus your mage with your priest with a little macro that reads (without the <>)

/target <mage name>
/focus

You use this macro one to 'focus' your mage with your priest and then you write your attack macros in the following way (inserting whatever spell you want at the end, the harm bit is there so it only tries to attack enemy targets, not friends. If you are doing a healing spell then use 'help' instead of harm):

/cast [target=focus,harm] Smite

Target:

You can have your priest target your mage with /target <mage name> and then write your macros to attack your target's target, that looks like this:
/cast [target=targettarget,harm] Smite

By character name:

You can write a macro to attack a particular characters target, you have to be grouped though for this to work (which you'd probably want to be grouped anyway).

/cast [target=<mage name>-target,harm] Smite

You can also have your alt assist your main, but I've not used this one, so don't want to give you the wrong information.