Welcome :)
If you start without the prefixes, you should write the code, generally, with all the semantically appropriate tags, first.
Then, you can decide what your goals are.
If you want W3C compliant CSS files, then you'd need to strip out the vendor specific prefixes by default. This would then allow the latest browsers to pick up the standardised CSS properties if they support them.
This will target less of your market than you might wish, so then you should ask if progressive enhancement is a possibility. If you can reasonably enhance the visuals by using css applied after the page has loaded, such as applying styles with jQuery, MooTools or Prototype libraries AND these libraries are already serving a purpose in your website, then you could apply additional styles with those libraries (and possibly use them in conjunction with Modernizr to determine which elements you may want to additionally support.
However, it's likely that a browser will skip a property it doesn't understand and will render the ones that it does, so I'd suggest that if you code it to W3C Standards first and then add in your additional vendor prefixes 'before' the final (correct) one, then you'll likely have satisfied reasonable measures to be compliant and meet design needs.
Edit:
There is a little bit of confusion between validation results from:
http://validator.w3.org/
and writing valid code related to vendor prefixes to get CSS effects cross-browser:
List of CSS vendor prefixes?
for working on some cross-browser CSS, you might find http://csspie.com, for IE compatibility with some CSS3 properties, useful along with http://www.colorzilla.com/gradient-editor/ for cross-browser gradients and http://cssplease.com for code that gives you alternative vendor prefixes, including different versions of IE support for many different properties.
In terms of validation, here's what W3C says about it: (see comments here: W3 VALID cross browser css gradient,) and official docs here: http://www.w3.org/TR/css3-syntax/#vendor-specific
If you code according to the specifications first and test your code out against that and then add in your vendor prefixes to get the same effects on the browsers you want to support (see progressive enhancement: What is Progressive Enhancement?) then you can be more confident that your code is supporting the appropriate standards, adding value to those with more advanced browsers and still useful for those without additional features (see also WAI III compliance, 508 compliance and others if you want to write a more inclusive site).
Caveat: Web Apps may not always be inclusive or follow these guidelines depending on who the audience is.