Q: Should my professor be the first author of my research?

While there are clear guidelines to decide who qualifies to be an author, deciding the sequence of authors is more of a grey area. The order of authorship can differ between fields or disciplines. In some fields, the names of authors are arranged alphabetically, so there is prcatically no importance on who is the first author. The ordering of authors also depends on the research group you are working with or the institution. In some institutions, it is the custom for the supervisor or PI to be the first author, although ideally that should not be the case. 

Ideally, the first author should be that person who has made the maximum intellectual contribution to work, in terms of designing the study, acquiring and analyzing data from experiments, and writing the manuscript. The order of authors should be decided by the relative overall contributions to the manuscript. It is common practice to have the supervisor or senior author as the last author, of course provided he/she, like all other authors, have met all criteria for authorship. Often, the supervisor or senior author is also the corresponding author, that is,  he/she is the one who receives all notifications from the journal. The group leader or a senior researcher is often the corresponding author because his/her contact address is not likely to change in the near future. In cases where the main contributor of the study is also the group leader, he or she can be both first and corresponding author for the study.

In your case, I feel, the professor could be the corresponding author, but you seem to be more deserving of first authorship.