@bexinfinity,
Without your mother's cooperation, this is going to be difficult.
I think before you go any further with this quest, you should sit down with your mother. Explain to her that it's not that you don't care about her feelings. It's that you are curious. Plus you have very real health reasons to want to know. After all, what if you're carrying a genetic disease? Such things are a sad fact of our lives and they are, at minimum, a reason to find out at least the basics.
If your mother refuses to listen, sit her down and be insistent. You can do so in a respectful manner. Explain that this is important to you, and that you are not doing this to try to give her any pain. You just want to know.
Be prepared for what she might say. There are any number of reasons why a mother would not want her child to search for a biological father. These include, but are not limited to:
- the child is the product of rape
- the child is the product of incest
- the father was married at the time of conception or has since wed
- the father is in prison
- the father is dead and did not, in the mother's opinion, live a good life
- the mother does not know who the father is, and is ashamed that there is more than one candidate or that the candidate(s) is/are unknown (e. g. a one-night stand, or a woman sleeping with two or more men or a child who is the product of group sex) or
- something I haven't thought of
But any of those could be the reasons why she has not been forthcoming with you. Listen and explain that you are not glorifying your biological father. You are not dissing her as a parent or any men in her life who have assumed the role of father (e. g. a grandfather, a stepfather, an uncle, or a trusted friend or long-time boyfriend, perhaps).
If your mother is still not on board with things, if you want to go ahead anyway, tell her you are doing so, in order that she won't be emotionally blindsided by whatever/whoever you might find.
And then - and
only then - talk to your grandparents, your aunts and uncles, and anyone who you know who was your mother's friend or even classmate around the time you were conceived. Ask open-ended questions - did my mom have a boyfriend? What was he like? Do you remember his full name? Do you know where he went to school, or worked, or lived? Again, you might not hear wonderful things.
And no matter what you hear or don't hear about your father, or if you decide to go through this at all, I hope you make it abundantly clear to your mother that you love her and think she did a great job raising you.