Thanks for the reply.

The context here is cryptography working in the multiplicative group of integers mod n. Im really just trying to get a basic grasp of group theory so I can understand the cryptography more.

I think it now makes sense why

isn't cyclic, because it doesn't have a prime order.

I was somewhat confused though because

this page says that the group should be cyclic if the modulus is of the form

where p is an odd prime, i still don't really understand why that fails to work with a modulus of 3^2?

As previously said, you're confusing stuff here: first, why did you add that * to ?? It doesn't appear at all in Wolfram's page and mathematicians understand that you're talking of the multiplicative group of units in the commutative finite ring of residues modulo , which is of order 6. The additive group of this ring, which is what Wolfram's talking about and denoting by , the hell knows why, IS cyclic of order 9, but it is ADDITIVE, not multiplicative!
If i can explain more on my context and where this is all coming from, part of a cryptosystem im looking at has this step:

So, I came to the conclusion that if a group is going to have multiple generators it a) needs to be cyclic and b) needs to have a prime order. Does that sound correct?

Not even close, I'm afraid to say. Plenty of groups have multiple generators and are not cyclic and, thus, neither of prime order. Take notice that there are cyclic groups of ANY finite order and also one (up to isomorphism) infinite cyclic group.
From here I went to trying the group

hoping the p^n (powers of odd primes) rule would work, but as you said it isnt of prime order, so every element wont be a generator.

Does that help explain where my mind is at? Thanks for any advice