Check out live.js:
Get familiar with them. I personally strongly prefer Chrome's developer tools over Firefox/Firebug, but regardless which one you choose, learn how to use the debugger.
As a seasoned programmer here are the steps you should follow for JS lib production.
If you're a Java developer, you may be familiar with jUnit. If so, qUnit may be more natural to you. Otherwise, I recommend you take a look at Jasmine or Mocha. While I prefer Mocha, my experience with Jasmine has generally been better for in-browser development thanks to the awesome Jasmine-Jquery plugin.
If you're writing qUnit tests, take a look at IntelliJ IDE which allows you to execute tests in addition to providing code coverage.
If you're developing on the browser, take a look at LiveReload. It'll watch files for you and auto-refresh your browser - great for instant feedback.
For browser compatibility, I would recommend you just get it to work on one first reasonably before worrying about the others. Check in from time to time to see if you find issues. See if jQuery can abstract some of that mess for you. Otherwise, take a look at Adobes BrowserLab.