Is it bad to start off the body paragraphs of an expository essay with "To begin with, secondly, thirdly..etc?"
I heard people say to use some sentances that connect to the previous paragraph instead of using firstly, secondly and thirdly for the opening of a sentance of the body paragraph in an essay
This depends to a great degree on your teacher and his/her preferences. Some teachers care more than others about stylistic things in expository essays.
For me, the most important thing is good organization. I want to see essays that have coherent paragraphs that each have a main idea and are all aimed at a central point. If an essay has those things, I do not care much about whether there are connections between paragraphs.
However, the very best essays will have good content, good organization, and style. Part of that style would be to tie paragraphs together with connecting sentences as much as possible.
Again, though, different teachers have different preferences with regard to this sort of thing. It is best to find out from each of the teachers for whom you must write essays.