Go to your local library or book store, and look for books that will help you to become familiar with German, Dutch, and Danish -- the three languages along with English that are most similar to Low German. Once you have at least a basic knowledge of all four languages, learning Low German will be a lot easier. You don't need to become fluent. Just look over the grammars and the lists of words to get a feel for them.
Use some of the resources below to work through some of the introductory lessons in Low German. They will give you a nice, easy way to become familiar with the basics of Low German grammar, as well as a little vocabulary. It is easy to get overwhelmed if you jump right into reading Low German, so take it slow.
Start listening to spoken Low German. You can start by listening to television and radio programs online. Because Low German is so distinct from High German, most of the media will be very clear about when they offer a program in Low German. A good place to start is Northern German Radio (Norddeutsche Rundfunk or NDR) which broadcasts from Hamburg, the center of the region where Low German is spoken by more than three million Germans, Dutch, and Danes. You are not trying to understand every word; you just want to get a general sense of the cadence and pronunciation of the dialect.
Make contact with Northern Germans through blogs and message boards. Many will be able to communicate in English. Let them know that you are learning Low German and that you are looking for resources and volunteers to help you with questions and translations. If you want to develop a steady group of teachers, make sure you offer them something in return. You might offer them help in learning English or in finding English-language websites that match their interests.
Start reading and translating Low German. You can find some magazines like Diesel (http://www.diesel-online.de) that are written in Low German, but most newspapers -- even in Hamburg and Bremen -- are written in High German. News is a good way to start your translating work, because they are usually stories you are familiar with from English language media, so there will be plenty of familiar referents. Try to avoid poetry at the beginning, because the word usage and grammar may be difficult and atypical. If you can't find Low German words in the limited dictionaries online, you may want to check the Mennonite dictionaries; the spellings may be different, but the pronunciation should be close enough to help sometimes. Use your e-mail and blogging buddies to help fill in gaps in your translations. And you can always check Danish, German, and Dutch dictionaries, because many Low German words have a cognate in one of the related languages.
The best way to learn Low German is to travel to Northern Germany and start getting to know the native speakers. You may eventually, through your online contacts, develop relationships strong enough to get you invited for a visit. There is no substitute for learning a language by immersing yourself among native speakers. Because even in the smaller towns children are raised to learn High German and English, you will have to be clear that you want to practice Low German, because the local residents may otherwise want to practice their English with you.