How do I update jQuery to the latest version I can download?
One way is to install the dev version of jQuery Update, which gives you the option to use jQuery 1.7 (by visiting the settings page at http://mysite.com/admin/config/development/jquery_update).

Beware though, currently it's known to break JavaScript elsewhere in the site (for example the widget dropdown on a Manage Fields page will no longer work). If you're prepared to fix up that, and any other small annoyances, this is probably your best bet. You should also note that using the dev version of a module isn't without risks, as the chances of it having bugs is a lot higher than using a specific release.

If you want to do it manually then you'll have a lot of core javascript code to rewrite to make your site work as expected; jQuery changed an awful lot between version 1.4/1.5 and 1.7 so a lot of the core Drupal javascript code simply won't work with it.

If you want to just drop in a version of jQuery 1.7 then you could probably implement hook_js_alter(), unset the core jQuery file, and add your own in it's place:

function MYMODULE_js_alter(&amp;$js) {
  $path = drupal_get_path('module', 'MYMODULE') . '/path/to/jquery.1.7.js';

  // Copy the current jQuery file settings and change
  $js[$path] = $js['misc/jquery.js'];

  // Update necessary settings
  $js[$path]['version'] = 1.7;
  $js[$path]['data'] = $path;

  // Finally remove the original jQuery
  unset($js['misc/jquery.js']);
}


Once you've done that, open up Developer Tools/Firebug and you'll see all the changes you're going to need to make to get the rest of the site JS to work :)

UPDATE

The latest version of jQuery update (7.x-2.3 at time of writing) will now allow you to update jQuery to v1.8. Very cool.

The jQuery Multi module is another option, thanks to goron for pointing that out in the comments. As he also notes, this module has a stable release and won't break any existing javascript.