I would second the Rubank series as well
However, there is one big pitfall awaiting the user of the Rubank Elementary Method, that being the fingering chart.
The large printed charts provided in the Elementary Method is a two sided affair. One side shows a "normal looking" clarinet, along with the long string of notes and appended fingerings. The other side shows a sickly looking horn, again with the notes and fingerings.
Many a student of the clarinet has been tripped up by this simple and obvious looking piece of material. If you use the Rubank, make sure to use the fingering chart with the anorexic looking instrument. Ignore the "normal looking" instrument version, as what is listed there are the fingerings for an Albert system horn.
The two instruments and note fingerings are similar enough to get things working, but working very wrong. The "first finger holes" are essentially reversed between the Albert and the Boehm (which is what you possess).
The "sickly looking" horn is a metal clarinet, a common enough instrument back in the day when the method was written by the wonderfully named Nilo Hovey, but relatively rare now. Sickly or not, metal Boehm horns finger the same as wooden (fat looking) ones. In the 1920's-1930's when he put his methods together, both metal clarinets (and Albert system instruments) would have been common enough for him not to omit either type from his work.
(And, it's not just the students that have had the problem. I've had parents intent upon helping their children take a look at the chart, seeing the similarities between the appearance of the Albert horn on the chart and the instrument in their child's hands, and get all insistent that they use that one for reference. I even went so far as to mark the chart for the Albert horn "Do Not Use", once I caught on to what was happening.)
And, when looking for a clarinet (or a saxophone, or (to a lesser extent) a bass clarinet) teacher, find one who is (ab initio) a clarinet (or a saxophone, or a bass clarinet) player. Many of us have been taught (by a proper teacher) one of the instruments, and then later picked up the other on our own. Better you should learn from someone who learned the tricks of the trade from someone who was taught them, rather than who picked them up on their own, since the self-taught approach very often misses out on the occasional nuance here and there.
(Clarinet and bass clarinet are obviously very similar, but there are differences in embouchure and the upper register that might escape some clarinet players who only "dabble" on bass.)
This is nothing against those who double (as a doubler myself, I can't cut my own throat) - it's just good advice.