It's a familiar enough story: parents, or the rabbi, tell the school committee that sixth-graders can't read Hebrew well enough to start b'nai mitzvah lessons. By "read Hebrew" they mean what Hebrew teachers usually call decoding: reading aloud with correct pronunciation but probably without comprehension.
Some Hebrew teachers are baffled by this statement. They point to the students' competence with the prescribed list of prayers, forgetting that the ability to read prayers does not always constitute reading. When you are my age and can't remember where you left your glasses, it is easy to forget how easy children memorize. In a class where there is frequent drill on specified prayers, many students achieve mastery of those prayers without developing the ability to read other Hebrew texts, even different prayers, with fluency. Since the student's Torah and Haftarah portions will be entirely new, and since they are unique to the student and won't be practiced over and over in class, the ability to decode somewhat fluently is essential.
The school's response is often to start teaching Hebrew earlier, but this doesn't solve the problem and may mask it further. Children who start learning to read Hebrew earlier don't necessarily achieve greater decoding proficiency; sometimes the result is worse. What children who start earlier, or who have more hours of class per week, do achieve is mastery of a longer list of prayers.
There's a reason for this. The methodology we use to teach Hebrew decoding in supplementary school builds on the students' secular-school language arts. Basically, we hope that students will have developed skill in phonetic reading in English that they can transfer to phonetic reading in Hebrew. (All Hebrew reading is phonetic, and students do better in Hebrew school if their secular school teaches reading through what used to be called "phonics." Nearly all schools do this now, because Federal education policy requires it, but a generation or so ago, lots of schools used non-phonetic methods.)
In my experience, the "primer year" is crucial. This is the year that students use a Hebrew primer that attempts to teach phonetic reading for mastery, teaching all the consonants and vowels and including polysyllabic words. Shalom Uvrachah, Ot la-ba'ot, Tiyulim, Likro u'Livarekh, and Z'man Likro are examples of Hebrew primers.
The publishers of all of these primers recommend them for grade 3 or grade 4. This is partly because all of them take for granted that students read at the third-grade level or higher. Some schools attempt to use one or another of these books in grade 2, but it doesn't result in better decoding overall, because only the second-graders who are reading English above grade level are ready for them.
Thus, I recommend making the primer year either grade 3 or grade 4. If your school meets two or more days a week, grade 3 is good. Not all of the students are firmly at third-grading reading level, but there is enough class time to work with those who aren't and who will need more help. I have come to agree with Dina Maiben that if your school meets only one day a week and has no more than one hour to devote to Hebrew, it is more productive to wait until grade 4 when nearly all students, except those with relevant learning disabilities, are reading at least at third-grade level.
Does this mean that you postpone all Hebrew until grade 3 or 4? In a word, no. Or, "lo!"
Think for a moment about how you learned to read English, if English is your native language. If you learned the sounds of the letters "d," "o," and "g" and then put them together to make dog, you experience a little thrill of revelation because you already knew the word dog and what it denoted.
Most of the time that doesn't happen in Hebrew school, because the primers teach mostly words that our students don't recognize. (In the first few weeks they teach a lot of nonsense words, too, but that's another problem.) It's the recognition of familiar words that cements the skill of phonetic reading, but in general our students don't know enough Hebrew words for this to work.
So what we should do in the early grades is teach words, not reading. That is, teach words orally, using "direct instruction" as much as possible. The means showing and demonstrating instead of telling and explaining, taking advantage of children's capacity for natural language acquisition. (People who cite natural language acquisition as a reason for starting Hebrew reading early overlook the fact that natural language acquisition is all about oral language, not about reading and writing.)
For example, instead of teaching that the letter shin has the sound of SH, teach that it is the initial sound in shofar. Do this the week before Rosh Hashanah and bring a shofar to class. If you can't blow it, arrange for a parent or synagogue volunteer to demonstrate. Let the children handle it and try to blow it. In subsequent weeks, review shofar (you can use a picture for the review). When you want to teach resh, have the rabbi visit and teach that in English we call her rabbi but in Hebrew the word is rav.
I like to use the Torah Aura series Now I Know My Alef Bet in grade 1; it teaches 50+ Hebrew words through their initial letters (it doesn't teach sofiot--final letters--or beged-kefet letters without dagesh). If you adopt it, be sure to use the vocabulary posters. In grade 2, my choice would be Journeys Through the Alef Bet, which uses the same approach but teaches all of the consonants. It adds one more word for each letter; you could also teach, as oral vocabulary, other words that will be in the primer the students eventually use.
For the primer year, I'm of six or seven minds. Many of the available primers have virtues and all have defects. Which book you use probably does not matter as much as how you use it: teach the students to associate the shape of a letter or vowel mark directly with its sound, bypassing the name of the letter and especially avoiding complex mnemonics. (Tav has the sound it does because it's tav, not because it has a toe sticking out.)
After the primer year, my preference is to emphasize modern Hebrew for comprehension; we use Behrman House's Shalom Ivrit. But if your school emphasizes prayer, at the very least teach all of the vocabulary and grammar that appear in the textbooks. The Hebrew series commonly used in grades 4 through 6 or 7, such as Hineni, S'fatai Tiftach, and Z'man Litfilah, all include some vocabulary and grammar, but teachers sometimes omit it in favor of prayer fluency, even though practicing prayers doesn't necessarily improve or even maintain students' decoding skills.
I spoke about this at CAJE in a session called "Hebrew Languish and How to Overcome It." The handout from that session, as a PDF, is here.