Learn Old English spelling and pronunciation. While the majority of Old English letters are the same as Modern English letters, there are a few characters you will need to learn. If you want to pronounce the language correctly, spend some time learning how to pronounce Old English vowels, which are markedly different than Modern English vowels. Much of the material available for learning Old English is geared toward scholars, so taking a peek at the International Phonetic Alphabet before attempting to learn the sounds would prove helpful.
Learn basic verb conjugations. In Modern English, verb endings are not very complex. For example, think of the verb "sing." In present tense we have two choices either "sing" or "sings." The verb only changes in third person singular. In Old English, on the other hand, verb suffixes display more variety. "I sing" would be "singe," while "you sing" would be "singest." "He sings" would be "singeþ," and all plural forms would be "singaþ." This is just an example of how much more complex the verb endings are.
Old English verbs come in two basic varieties: weak verbs and strongs verbs. There are multiple sub-classes for each type that display different conjugation patterns. There are several websites that offer material for learning Old English verbs and grammar. One of the most comprehensive resources is the ORB: The Online Reference Book for Medieval Studies. Indexed there is an introductory course in Old English by professor Murray McGillivary of the University of Calgary.
Develop your vocabulary. This is where a good Old English dictionary comes in. For the beginning student, an affordable option is Wordcraft's New English to Old English Dictionary and Thesaurus. McGillivray's course offers vocabulary lessons and flashcards as part of the online resource materials. Alternatively you could buy a textbook like Teach Yourself Old English (includes 2 CDs) or Peter S. Baker's Introduction to Old English.
Practice reading shorter passages. It may be tempting to jump straight to Beowulf, but developing your reading can take some time. Start with shorter passages like poems or passages from the Old English Bible. The ORB page lists links for several short texts that you can read for practice. If you are familiar with the Bible, choosing passages from the Old English version would be a great learning tool. You may be able to find some Old English readers in a nearby research library, but if you are studying at home you can order a copy of something like Sweet's Old English Reader to get you started.