Depends on the language. For Mandarin Chinese, there are (free online on edx.org) courses based on pinyin, ignoring the Chinese symbols completely.
Of course being able to speak, but not being able to read, will severely limit your learning potential - but if might be a valid first step in learning Mandarin. Luckily, some/most Chinese (I heard - I have no personal experience with Chinese) will be able to understand your written pinyin, and there even might be pinyin-to-mandarin translator programs.
For most other languages, and especially for languages with nearly-phonetic script like Russian or Thai, learning the alphabet first is fully warranted, and postponing it does not make sense (if you seriously intend to learn language beyond few tourist phrases using some local romanization system).
I also read that to learn Japanese, it is preferable to learn one of the simple alphabets (I forgot which one), because (Japanese learners said) romanization will teach you wrong habits.