In my experience, comparing yourself to others with regard to your skating progress is not helpful. It might work for some people if they are spurred on to greater efforts by feeling like they have to compete with others, but I think it probably serves to distract and demoralise more than it helps. I think the same applies to setting yourself arbitrary, time-bound goals or having time-bound expectations of your achievement. Everyone learns at a different pace, and you are at a very early point in your skating journey. I think you should focus on trying to do what the coach showed you, and on enjoying the process.
Will you be having follow up lessons? I think it would be helpful as the coach will be able to tell you whether what you have been practicing is correct. Often people think they are doing something when they are really not.
As for people laughing at you, I suppose it's possible this is the case - if so that says everything about the people laughing and nothing about you - but in my experience people are rarely paying any attention to what others are doing.
Having decent skates certainly makes the experience more pleasant and a bit easier, so if you can afford it and you intend to give skating a go for at least a few months then a modest investment in some reasonable quality entry level skates seems like an idea worth considering. If you decide skating is not for you, you could probably get a portion of the money spent back as there is a second hand market for skates in reasonable condition. If you're having regular lessons then the money spent on new skates will likely be a good deal less than the total you spend on lessons.
Happy skating!
