<?php
class Rkt_Breadcrumbs_Block_Breadcrumbs extends Mage_Catalog_Block_Breadcrumbs
{
    protected function _prepareLayout()
    {
        if ($breadcrumbsBlock = $this->getLayout()->getBlock('breadcrumbs')) {
            
            $title = array();
            $path  = array_shift(Mage::helper('catalog')->getBreadcrumbPath());

            foreach ($path as $name => $breadcrumb) {
                $breadcrumbsBlock->addCrumb($name, $breadcrumb);
                $title[] = $breadcrumb['label'];
            }

            if ($headBlock = $this->getLayout()->getBlock('head')) {
                $headBlock->setTitle(join($this->getTitleSeparator(), array_reverse($title)));
            }
        }
        return parent::_prepareLayout();
    }
}

<?php
class Rkt_Breadcrumbs_Block_Breadcrumbs extends Mage_Catalog_Block_Breadcrumbs
{
    protected function _prepareLayout()
    {
        if ($breadcrumbsBlock = $this->getLayout()->getBlock('breadcrumbs')) {
            
            $title = array();
            $path  = Mage::helper('catalog')->getBreadcrumbPath();
            array_shift($path);
            foreach ($path as $name => $breadcrumb) {
                $breadcrumbsBlock->addCrumb($name, $breadcrumb);
                $title[] = $breadcrumb['label'];
            }

            if ($headBlock = $this->getLayout()->getBlock('head')) {
                $headBlock->setTitle(join($this->getTitleSeparator(), array_reverse($title)));
            }
        }
        return parent::_prepareLayout();
    }
}

EDIT

This job is somewhat difficult to achieve. This is because, by default magento adds home breadcrumb and category breadcrumbs from two different files. So basically you cannot avoid a rewrite atleast. So make your work more clean, you need to create a module. I am going to call this module Rkt_Breadcrumbs. First create config.xml for your module.

app\code\local\Rkt/Breadcrumbs/etc/config.xml

<config>
    <modules>
        <Rkt_Breadcrumbs>
            <version>1.0.0</version>
        </Rkt_Breadcrumbs>
    </modules>
    <global>
        <blocks>
            <catalog>
                <rewrite>
                    <breadcrumbs>Rkt_Breadcrumbs_Block_Breadcrumbs</breadcrumbs>
                </rewrite>
            </catalog>
        </blocks>
    </global>
</config>

So our module just tells to magento that, it needs to rewrite the class Mage_Catalog_Block_Breadcrumbs with our class Rkt_Breadcrumbs_Block_Breadcrumbs. Now we can define our rewrite class

File : app\code\local\Rkt/Breadcrumbs/Block/Breadcrumbs.php

<?php
class Rkt_Breadcrumbs_Block_Breadcrumbs extends Mage_Catalog_Block_Breadcrumbs
{
    protected function _prepareLayout()
    {
        if ($breadcrumbsBlock = $this->getLayout()->getBlock('breadcrumbs')) {
            
            $title = array();
            $path  = array_shift(Mage::helper('catalog')->getBreadcrumbPath());

            foreach ($path as $name => $breadcrumb) {
                $breadcrumbsBlock->addCrumb($name, $breadcrumb);
                $title[] = $breadcrumb['label'];
            }

            if ($headBlock = $this->getLayout()->getBlock('head')) {
                $headBlock->setTitle(join($this->getTitleSeparator(), array_reverse($title)));
            }
        }
        return parent::_prepareLayout();
    }
}

Here we are rewriting the method _prepareLayout(). Here if you compare the two methods, you can see that our new _prepareLayout() method do two jobs.

1. It removes Home breadcrumb

2. It removes root category from the category array which is generated via the helper class Mage_Catalog_Helper_Data::getBreadcrumbPath() using the method array_shift. That's it. You are done. Dont forget to add the activation file of the module. It looks like this.

File : app\etc\modules/Rkt_Breadcrumbs.xml

<config>
    <modules>
        <Rkt_Breadcrumbs>
            <active>true</active>
            <codePool>local</codePool>
            <depends> 
                <Mage_Catalog/>
            </depends>
        </Rkt_Breadcrumbs>
    </modules>
</config>

Note : This will resolve issues in category page only.