I would like to show the following links (I think they are called badges) on my homepage.
Assume my website is called 'abc'
An example of this can be seen on startups.com
I am using PHP for development - I would like to learn how to put the Facebook and Twitter links on my homepage.
Facebook Fan Box is exactly what you're talking about
Most people just link a promo image to their Twitter profile. Some free ones here that I've seen a couple sites use.
do you want to automatically Tweet out new posts? Are you using any kind of CMS for which a plugin might be available? Otherwise there are a ton of PHP Twitter libraries out there you can hook into the publish action on your site (assuming you have a custom CMS of some kind). Or if you have an RSS feed setup, you can have Twitterfeed Tweet it out.
If you just want visitors to be able to easily Tweet out your links, Tweetmeme (among lots of others, but it's the service I'm most familiar with) has a nice Retweet button.