putting songs purchased in iTunes on to my iPhone 5

I'm not sure I understand the term syncing. I would like to get my purchased songs that are on my computer on to my new iPhone 5. I do have an iPod that I sync with my computer. In apple support, I followed the directions for syncing my iPhone 5 with iTunes on my computer.

In fact I just now answered a question, or raised another one. The iTunes program on my computer has Store and under that iTunes Store and Purchased. Under than there is my Device with "purchased" under this tab there are 23 songs. Those were downloaded without syncing. I start pressing buttons and god only knows what I get started. I think I downloaded songs over the ATT network. That was arduously slow, so I went to syncing with my computer, i.e. connected to my computer and started the syncing process.

Did I accidentally re-purchase 23 songs?