I get this question fairly often from new students. I tell them an answer that may sound cocksure, but I think it’s realistic. I tell them that after two or three lessons they should be able to tell if my way of teaching has something to offer them and they should be able to notice improvements in the first few weeks, not months or years.
In order to select criteria to use to judge whether you are improving, it is important to be clear about your goals. Since I am a functional teacher who likes to help with technical issues (which are not 100% separable from musical issues, since function is measured by musical tasks), I like to suggest that functional criteria be used, along with some subjective feelings. However, I also respect non-technical goals such as how to learn songs, how to sing more confidently, and how to audition for shows. These are also helped by functional study but the student will have to judge for themselves whether they feel better about what they’re doing.
Some functional criteria – Can your voice do more, and more easily than before? That one sentence sums it up, as far as the technique goes. Are you more comfortable in the extremes of your pitch range, do you have more options about dynamics, can you make clear vowels and form words in a way that works with your singing rather than against it? Does your voice move better? Are you singing in tune? How does it feel to you? Is it easier? Is it pleasurable? Do you feel more confident? Do you like your singing more than you did before? Are you making a type and size of sound that is usable in the genre of music that you sing?
After two or three months with a new teacher, some of these questions should be answered “yes”.