Both of these services use short status updates which can contain your message and a link to a related web page. If someone "likes" your Facebook page then your status updates will appear in their "most recent" or "top news" feeds. The position of it on the list may change if people are commenting on it. Twitter is slightly more haphazard as your update will appear chronologically in among the updates of everyone else your follower is following, but it is still a very good communication tool.
A personal user of Facebook and Twitter would probably not want the same status updates going to each feed, because a personal Facebook page is restricted to friends only with Twitter usually being public. A business, on the other hand, would probably want both accounts carrying the same news so there is an easy way to achieve this.
1. Create a Facebook page for your business (not a group).
2. Create a Twitter profile.
3. Go to http://www.facebook.com/twitter/ and connect the two accounts.
Once this has been done any status updates you make to your Facebook page will automatically be sent out on Twitter with a shortened link to the original page on Facebook.
Be prepared to interact with users on both services by regularly checking and engaging in discussions and comment threads on Facebook.
Twitter has a number of third party automation tools which you will want to start using once you get more involved with it. One of these is www.socialoomph.com which can do a number of clever things including sending a personal message autoresponse to anyone who follows you and automatically follow them back. This can allow you to grow your network without having to manually check and follow each new follower. Socialoomph can also be used to trickle feed updates automatically at certain times.
If you have a blog then www.twitterfeed.com can send an update to twitter each time you add a new article. This can be useful in retaining engagement with customers.
This is just the start, but remember it is about building and engaging with a community of people who have similar interests to you. Whatever updates you send out, my advice would be to limit it to one in five of a commercial nature. The rest should be informational or people will just start ignoring you or not feel good about your organisation.