Firstly, it would be worth going over why you should try it in the first place. I've found it's great for boosting traffic, but even more so for keeping your finger on the pulse of what interests an online audience - for newsgathering, and just reminding people not only that you're here, but that you're part of the community - and drawing them into it too.
So if you want a go, here's some advice.
Before you start, it's essential to get yourself a good avatar and bio so people know who you are, what you intend doing on Twitter, and hopefully that you're going to be interesting enough to follow back.
Follow as many local tweeple as you can
Use the Find People search in Twitter, which allows you to search by location
Set up an RSS feed via www.search.twitter.com for placenames to see who's tweeting about your area.
Find established accounts (I had @brightonfeed, @brightonhovecc) and follow their local followers
Look out for lists of people living in your area like this one (tip - people's profiles show both which lists they've created, and which lists they're on). You can find more on Listorious, although it's far from comprehensive
Use Trendsmap, if it has a page for your neighbourhood
There are other applications like Twitterlocal, but I've not found them that useful as they're not as accurate as you'd like in identifying where people actually are
Every time you follow someone, they get an alert telling them, so hopefully a good few of them will follow you back. But beware following too many people at once - you're less likely to be followed if you're following far more people than you have following you.
I won't go through the nuts and bolts of how Twitter works - loads has been written on that already. I found Shane Richmond's post here useful when I started out (last updated Jan 2009). Mashable also has a great range of articles on all aspects of Twitter.
Good things to tweet are news and newsroom insights. If your paper doesn't already have an account using Twitterfeed to autopublish your stories, set one up as it's the quickest way to give people news by retweeting. People love insights into how their news is put together. so you can probably be more naval-gazing than you think (my second-most popular link was our style guide!). But be very careful not to inadvertently mock contacts, use any trace of gallows humour, or give away exclusives.
@reply to people. Nobody expects you to sit on Twitter all day, but when you do log in, chances are someone will have said something interesting you can respond to. You'll also find lots of local interest tweets which cry out for a response from your RSS feed of local mentions. Don't worry about not knowing the person who's tweeted - they won't mind.
RT (retweet) people. First of all, it will make the person you RT feel interesting (and if you're RTing them, they are). Secondly, it's an easy way of being interesting yourself. Lastly, it shows that you're engaged in what's going on, and not just in broadcasting your own links.
Crack (clean) jokes. Try and be witty. Channel your inner Dorothy Parker. And avoid tweeting about your lunch or anything personal that a stranger wouldn't find funny or interesting (although having said that, it's often the more personal tweets which get the biggest response).
Follow the principles of a good community manager, even though it's not your community you're managing. Be consistent and firm, but responsive and ready to apologise if you've messed up - which, at some point, you inevitably will. But if you say sorry, chances are most people will respect you more afterwards than they did before.
Consider using hashtags. We use two - #bricom as part of our Save Our Service campaign, and #brightoh to help compile our Overheard in Brighton blog.
Crowdsource. If you need general case studies, Twitter's a great bet. the other day, it even managed to track down a woman who walks a ferret around Brighton in under an hour.
Take it into the real world. If there are local tweet-ups, go to them. If there aren't, organise one. Twestival is a great thing to get involved in.
Any more suggestions?