This must be split owing to the lisserv's limits. If you don't like this part, skip the next!
Let me say at the top: Old Irish is a bear of a language. It is the hardest IE lg I have ever attempted and is rivaled only by Navajo in its overall, interlinked complexity of phonology, morphology, and syntax. (Navajo semantics is even harder, however.) BUT it is learnable and in the end very rewarding, both for the sheer accomplishment of coming to grips with its structure and for the richness and beauty of the literature it transmits. Below is fairly long (sorry for those who are not neophytes-I mean no insult to you by being very basic here for true novices) but I hope this exposition will be encouraging for the long run. Because my initial recommendation, "to immerse in the texts" was given out of context and that context is critical, I have addressed this answer widely because learning the lg is a multi-dimensional matter. If this seems self-indulgent, forgive me.
A lot depends on whether you want to do this on your own or are in class or group with the guidance of someone reasonably knowledgeable and infinitely supportive. Unfortunately for the beginner, OIr is a frustratingly complex lg. Since the orthography is not directly phonetic, even looking up a given word in a general glossary may be futile. Who would predict that the negative of "asbeir" would be 'ní epur"? This sense of frustration is increased if the student is already experienced with learning more tractable languages such as Gothic or Greek! So we cannot count on using the traditional progression from the "powers of the letters" (orthography) to simple morphology to a criminally simplified syntax, which order we owe to Dionysus of Thrax from about 100 BCE. OIr apparently didn't get the announcement about the prescribed drill.
Instead, in order to process the orthography the student must learn a fair amount of syntax and morphology in order to understand the realization of the phonological processes called mutations (conditioned by an historical but perhaps no longer present preceding vowel or nasal: a tíg 'their house' a thíg 'his house' a ntíg 'their house' ). (And, of course, following these mutations is further complicated by the fact that OIr orthography is partially ambiguous, so one spelling is each case can have two meanings. Sigh....) In other words, one must learn several fairly complex things at the same time, which is fine if one is either very disciplined and self-confident, or has continual reassurance and guidance from a supportive teacher.