My street car issues....
Congrats on the license and the car! I had the pleasure of running with your dad out at Nelson Ledges last year...well I was in the same run group, can't really say I was running with him
Anyway my 83 street car had a rusty tank. It got to the point I had to change the filter every 2 months. (The stock filters are really small). As people have already warned you: make sure you have good fuel levels in the bowls even if you have already checked it before.
What is the lowest RPM it will run?
I would start by setting the idle mixture screw to the initial position. Do you have a Mazda manual or Haynes? After that adjust the throttle stop.
There are about 4 miles of vacuum lines in the emissions systems. A small crack can easily mess you up. If you so start replacing them be very, very careful removing them from the plastic solenoids. Those plastic valve bodies are 26 years old and very brittle now. I slit my hoses with a razor before pulling them off and replacing them. (well after I broke half of them I started slitting the hoses).
There are also several big valves on the outside of the carb that work with the air control valve. Checking them is time consuming, but pretty straight forward. I am not a huge fan of the Haynes books, but they are pretty good at describing how to check that whole mess out.
If all the emissions stuff checks out, then check the timing and vacuum advance.
If it still does not work then it could be carb rebuild time. I messed two up before I got it right.
Moron move #1: do not wait for your mom to leave the house and then put the carb body and horn in the dishwasher to clean them! While it cleans them very, very well, the dish washer detergent is corrosive to the aluminum and pits the sealing surfaces.
(okay, hope everyone had a good laugh and no one peed their pants...)
Moron move #2: there are some very small balls inside the carb. I can't remember exactly where, but they are very easy lose when disassembling a dirty old carb. Work on a big clean surface, take pictures and make sure you can find small parts that you drop.
Good Luck!