you will need to edit style.css file.
to move it to right, place this within style.css file:
.search { float: right; }
as for styling, there are a zillion tutorials on the net.
remember that your search form is contained with a search class, so use that. your code:
<li class="search">
<form method="get" id="searchform" action="/" abineguid="76E467BE146545918F6546ACA2302728"><label for="s" class="assistive-text">Search</label><input type="text" class="field" name="s" id="s" placeholder="Search"><input type="submit" class="submit" name="submit" id="searchsubmit" value="Search"></form>
</li>
for example, to change background of input field, use:
.search input { background-color: #000; }
hope you got the idea.
ensure you take proper backups and work on a child theme for safety reasons.
