Why Learn Hebrew?
Hebrew is the language of Tanakh, of the siddur, of three thousand years of Jewish thought, and (since 1948) of the modern State of Israel. Some Hebrew is a necessity. More than some, a deep enrichment.
Summary. A New Jew needs enough Hebrew to follow the siddur, recite the central prayers, and read the Torah text alongside translation. This is a small fraction of what fluent Hebrew opens up — the Talmud, Midrash, classical commentaries, and modern Israeli literature all reward the learner who continues. The aleph-bet itself can be learned in a week; reading with vowels in a month; reading without vowels and a working vocabulary in a year of consistent study. The investment pays dividends for life.
What You Need
For prayer participation: the ability to read Hebrew with vowels (nikkud) at moderate speed. The siddur is fully vocalized; most prayers repeat from service to service; with practice the eye picks up the patterns and the mouth follows the eye. This is the minimum and is achievable by any motivated adult in a few months of practice.
For Torah reading: the ability to read Hebrew without vowels (the standard Torah scroll has only consonants), at least with familiar passages. This is harder; most adult learners reach passable competence after a year or more of consistent practice.
For learning Mishnah, Talmud, Midrash, and commentaries: vocabulary, grammar, and idiom build over years. The Mishnah is in Mishnaic Hebrew; the Talmud is largely Aramaic (with Hebrew quotations); Rashi is in medieval Hebrew with distinctive abbreviations; the Tanakh is in Biblical Hebrew. Each is a related but distinct register. A New Jew should aim for the Mishnaic and Biblical first; Aramaic comes later.
Three Hebrews
Biblical Hebrew — the language of Tanakh, formal and poetic, with a relatively small vocabulary (~8,000 distinct words). The grammatical structure is somewhat different from modern Hebrew.
Rabbinic / Mishnaic Hebrew — the language of the Mishnah and Midrash, more concise and conversational than Biblical, with Aramaic loanwords and a different verb system.
Modern Hebrew (Ivrit) — the revived language of the State of Israel, drawing heavily on Biblical and Mishnaic vocabulary but with modern grammar and the addition of thousands of neologisms and loanwords.
All three are written in the same alphabet (the square Aramaic script standard since the Babylonian Exile, alongside Rashi script and cursive). A learner of any one gains substantial leverage in the others.
Sources
Rambam, Hilchot Tefillah 12 — on learning to read Torah aloud.
Mishnah Berachot 2:3 — on the language of the Shema.
Further Reading
George Schoolfield, Learn to Read Hebrew in 6 Weeks.
Page Kelley, Biblical Hebrew: An Introductory Grammar.
Ross & Glinert, Modern Hebrew: An Essential Grammar.