I can't put music on my iPhone?

I use to have all my music on my parents computer but I got a new lap top and iPhone. I was having trouble getting music onto my iPhone from my old computer but somehow I managed to get some of my favorite songs onto my iPhone, but I couldn't get anything on to my laptop. NOW I have some music on my iTunes on my new lap top that I want to transfer over to my iPhone but it says that I have to erase and sync my music from my lap top, but if I do that I lose all my other music that I have from my old computer. Is there anyway around this?

"I couldn't get anything on to my laptop."

From the iPhone? Why would you think you could. Apple wants to sell music and they know that that's going to be harder if people could just connect iPhones or iPods to computers and copy from them.

Here's a question - if the music from the old computer was purchased on iTunes, you should be able to copy them from the iTunes store using "transfer purchases" - did you try that? If the songs were not purchased from iTunes, were you somehow unable to copy them to flash drive or CD. Even if it broke down, you would still have saved most of your songs if you routinely backed up your files. Your question says nothing about either of these things.

"I want to transfer over to my iPhone but it says that I have to erase and sync my music from my lap top"

Sounds like a great reason NOT to sync. I've put thousands of songs on my iPod and never synced it once, Instead I use manual management. It works on old devices like the iPod Touch 3 & 4, as well as iPhone 4 & 5, and iPad 1 & 3 - so anybody who says it doesn't is just ignorant.