Can I get to my purchased songs from another computer???

Depending upon which country that you are in then you might be able to re-download them from the store for free via the Purchased link under Quick Links on the right-hand side of the iTunes store home page on your computer's iTunes : Re-downloading.

Or you can copy them from your current computer to it : copying music between computers.

As King Penguin alluded to earlier:

Downloading past purchases from the App Store, iBookstore, and iTunes Store - http://support.apple.com/kb/ht2519 - enabled with iTunes 10.3 and newer; not available in all countries; apps, books (not audiobooks), music, t.v. shows, and movies (some - not all studios have permitted this). Movies currently available in the USA only. Downloading previously purchased movies and TV shows requires iTunes 10.6 or later. Discontinued items not available. For items not included in the iCloud list (e.g., ringtones), or locations or computer systems where iCloud is not (yet?) available, you only get one download per fee paid. Apple notes it is your responsibility to back up your purchases.

Select the store on the left side of iTunes.
Click on Purchased on the right side under Quick Links.
You can re-download your available previous purchases.

It would help to know in what country you are located. In some countries you can download again for free, in ohters not. See the extra notes I included above.

Follow the directions here:

Downloading past purchases from the App Store, iBookstore, and iTunes Store

It is free.

Just so you know, that's the same document to which both King Penguin and I referred you earlier.

It seems repitition is the key.