if you just want to remove breadcrumbs from home page and category pages, then you can do it easily by a layout update. Use `local.xml` file to do this.

File : `app\design\frontend\<your_package>\<your_theme>\layout\local.xml`

    <?xml version="1.0"?>
    <layout version="0.1.0">
        <default>
            <reference name="root">
               <remove name="breadcrumbs"/>
            </reference>
        </default>

        <catalog_category_default>
            <reference name="root">
               <remove name="breadcrumbs"/>
            </reference>
        </catalog_category_default>

        <catalog_category_layered>
            <reference name="root">
               <remove name="breadcrumbs"/>
            </reference>
        </catalog_category_layered>
     </layout>


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();
                $path  = Mage::helper('catalog')->getBreadcrumbPath();
                array_shift($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();
        }
    }

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.