Where do I go on Adobe.com to download and install the products I own on a new computer? I can find serial numbers, etc., but not the download spot.
Since you didn't tell us what Adobe products you have, I can only give a general answer.
Go to the home page for that product, download the free trial, install the free trial and apply your serial number
From the main Adobe page, you would only be able to download CS6, not CS5.
However, you should be able to download CS5 from your download history. Suggestion: save the downloaded installer file from Adobe on your hard disk somewhere, so you can always get it.