Currently I'm using a custom (rather ugly) image:
Is there something more standardized?
Can I show how many people follow me on Twitter, just as the 'Like' button shows how many people like my site?
A "Like" on Facebook is usually is a "Star" (or "add to favorites") on Twitter. For an overview of tweets that received stars, Favstar.fm seems to be the most common site. As long as it is about single tweets the star seems to be the established symbol of "I like".
When it comes to just linking to a related Twitter account, I guess there are multiple button-like images that Twitter provides for linking.
Like this one , which looks similar to your custom one.
I hope this is helpful, otherwise it seems I did not fully understand what you were looking for.
Twitter now officially offers a Tweet button which shows how many people have tweeted the URL of the current page it's on (or another page if your customise it).