I have a site that is already indexed by Google, but as I posted to the address when the site was still being tested, Google indexed some results like broken URLs, test pages, wrong descriptions, etc.
Do I need to create a new domain to index everything correctly, or is there any manual way to re-index that site?
Google automatically re-indexes the site. There are Google tools for this. Take a look at this question about this: How to remove a site from Google?
Apart from that I would say google likes stability. A too-new domain will not get much attention from Google, so changing domain too easily is bad.
What you should do is create a site map and make redirects of pages that do not exist.
About sitemap, it is a sound file name sitemap.xml that should be in the site with information about the site links. You can automatically generate one here: link and then edit to fix details.
Google has a very informative sitemap page: link
Here is an example of the syntax:
<?xml version="1.0" encoding="UTF-8"?>
<urlset xmlns="http://www.sitemaps.org/schemas/sitemap/0.9"
xmlns:image="http://www.google.com/schemas/sitemap-image/1.1"
xmlns:video="http://www.google.com/schemas/sitemap-video/1.1">
<url>
<loc>http://www.example.com/foo.html</loc>
<image:image>
<image:loc>http://example.com/image.jpg</image:loc>
</image:image>
<video:video>
<video:content_loc>
http://www.example.com/video123.flv
</video:content_loc>
<video:player_loc allow_embed="yes" autoplay="ap=1">
http://www.example.com/videoplayer.swf?video=123
</video:player_loc>
<video:thumbnail_loc>
http://www.example.com/thumbs/123.jpg
</video:thumbnail_loc>
<video:title>Grilling steaks for summer</video:title>
<video:description>
Get perfectly done steaks every time
</video:description>
</video:video>
</url>
</urlset>