Well, start by repairing permissions from Disk Utility if possible. It's pretty easy, you just select your hard drive and press repair permissions. Disk Utility is in the Utilities folder.
Then, Get Info on the Applications folder via the File menu and make sure your privileges are correct. You may need to unlock the settings though.
Then permissions and ownership of the applications directory are not your issue.
Do you have another account that is an admin? You have to have at least one admin account on the computer...hmmm. It shouldn't let there be less than one.
P.S. Now that I think about it. How do you know your apps folder is read-only? Just out of curiosity, I want to make sure I really understand the issue you have.