I would like to point out that the English translation isn't 100% OK if we want to be 100% accurate. I put "I don't smoke cigarettes" which was (for some reason) marked as incorrect. If "dohányzik" means "to smoke" (in general) and "cigarettázik" "to smoke cigarettes" (something specific), then the correct English translation should be "I don't smoke cigarettes". Don't really want to split hairs here, just wanted to draw attention. :-)
It would actually be more appropriate to say "cigarettázom" because this is what is called an -ik verb. Basically, if a verb ends in a z like this one, it conjugates slightly differently. 1st person singular is always -om/-em and 3rd person singular is -ik. However, Hatcher is right that people don't always strictly follow that rule.
The "ikes" paradigm is a historical remnant that is fading out of the language. You can still find some forms of it (limited to first person), alongside the non-ikes form, like "eszem," "alszom," "dolgozom," "fekszem". There is a stylistic difference; the -m form usually sounds more sophisticated, but it is not a rule. It is not any more grammatically correct than the -k equivalent. Only prescriptivists cling to the -m form and claim that "eszek" and "dolgozok" are "wrong" or that speakers who use them are "destroying" the language. But that misses the bigger picture. Languages change, and the ikes paradigm has long been on its way out. For example, here are some other forms from the ikes paradigm, which no one would use today or claim are more "correct": én aludnám, ő aludnék. If you read Hungarian, this is a good article about it: http://helyesiras.mta.hu/helyesiras/blog/show/maradvany-ujdonsag-ikes-ragozas
I also want to point out that not all "-zik" verbs are conjugated according to the ikes paradigm. Many people say "dolgozom," but almost nobody says "vérzem."
Some words that can go both ways are more likely to follow the ikes pattern in everyday usage, and others are more likely to follow the non-ikes pattern. For example, even Google will correct you if you type "igyekszek," since "igyekszem" is so much more frequently used. Both are grammatical; it's just a matter of style.
Thank you for your comment and especially for the link. I'm not a native speaker of Hungarian, but I've lived there and worked in the language for many years. However, one disadvantage that does still give me is that I often speak a little bit too "by the book". I know that languages change and are always changing, but with Hungarian I kind of look in as the outsider trying to follow the change, but unable to actually participate in it if that makes sense. Your article was very interesting though - I hadn't realized that -ik verbs were originally reflexive and passive only. That actually makes a lot of sense though. I had always assumed - based on how I was taught - that it was a phonetic rule. Thank you so much for your comments! I do wonder now though what they will do here. Should I write assuming the historical -ik conjugation is correct? Or will I need to be able to know what is most common these days? Or do you think that both conjugations for all -ik verbs should be accepted?