I don't what your doctor will recommend but the recommendation will undoubtedly be governed largely the stage of the cancer. Which is determined by several variables, including size of the tumor, whether any lymph nodes are affected and other more qualitative things like location of the cancer and containment of cancer cells. The other factors include the type of cancer which the biopsy will note.

I'm assuming, since you have a diagnosis, that there was at least a biopsy done and the pathology report will yield a lot of information. I'd start by asking whoever made the cancer diagnosis these question. (DCIS is a cancer diagnosis though might be a Stage 0 which some oncologists consider a kind of 'pre-cancer' or possible-to-become-cancer.) And get a copy of any medical record in case you want to get a second opinion. And ask about any testing that can yield more data.

If a lumpectomy is suggested and you're worried about having one, I had one and it was a very simple procedure that healed, in my case at least, scar-free. (A small tumor, excellent surgeon, cancer-free margins.) I have a small scar from the removal of a lymph node that looks like a kitten-scratch and is nearly invisible eleven months later.) I had no pain from the lumpectomy and lymph node removal after discomfort (buy not pain) the first day.