Help explain multi alleles in terms of actual genetic code
I understand that someones genetic makeup at an allele is usually
  denoted as (AA,Aa,aa).


No. That's what you learn in high school, because it's easy, and because it's what Mendel worked out for the 7 traits he studied in pea plants.  Real life is a lot more complex.  Most phenotypic differences cannot be explained in terms of exactly two alleles, one of which is classically dominant to the other.


  That means that instead of an A-T pair you get a C-G pair on none, 1
  or 2 copies.


Not necessarily.  Two alleles might differ at many many points in their sequence.

And you seem to be a bit confused about how people write out DNA sequence.  Even though DNA is a double helix, when we talk about what the sequence is, we refer only to one strand, like this

GAGCCATGGGGCGCCATGTTGCCACCAGCTGCCACGTGGCCTGGCTTTTGGTGCTGATCT

A mutation might turn that last T into an A, but we don't say "oh, nothing happened, it's still an A-T pair" because it matters which strand has which letter.


  Now, what exactly happens when you have a tri-allele (as in blood
  type).


Well, blood type is more complicated that just ABO...but here's a pretty old paper with some helpful details

http://www.ncbi.nlm.nih.gov/pubmed/2333095

"Here we present a molecular basis for the ABO genotypes. The A and B genes differ in a few single-base substitutions, changing four amino-acid residues that may cause differences in A and B transferase specificity.  A critical single-base deletion was found in the O gene, which results in an entirely different, inactive protein incapable of modifying the H antigen."