My daughter came out as trans almost exactly a year ago. She was 17, so slightly older--those two years mean a lot.
In the run-up to the announcement, our daughter experimented with dress and nail polish and growing out her hair. We complimented her hair, bought her nail polish, and complimented her clothes (usually the patterns because her style didn't make much sense to us) every morning that it made sense. She had announced at 14 that she was bi, so we expected her to "carve her own road" somewhat in the style department.
When our daughter came out at 17, we managed to keep our cool in the moment and react supportively. (There is nothing that I am more proud of myself for because it was the most critical moment in my child's life.) Because she was older, I put the ball in motion immediately. I located the adolescent gender clinic at our local children's hospital and made a soonest-possible appointment. I figured that professionals, who see this regularly, were the best people to help my daughter sort out her desire for hormone-affirming therapy and tell us if this were "real" or a "phase." This might seem counter-intuitive, but as I knew NOTHING about feeling transgender, I was at a loss for how to counsel my child on my own.
They don't dole out the hormones on the first visit. They do a battery of blood tests to ensure good health. They require the affirmation of a licensed mental health professional who has seen the child in person. Also, the appointments are part provider and child one-on-one then part family together with the provider. At 15 your child is accorded some medical privacy, I think per HIPAA, so be prepared for that. It's hard to let go.
She benefits tremendously from seeing a therapist who is herself somewhat gender fluid. I cannot recommend strongly enough that you try to find a therapist for your child who is themselves queer in some respect. They will provide support and help clarify feelings and thoughts in ways you cannot possibly hope to, even though you love and support your child and want the best.
We also benefited from therapy for ourselves (my husband and myself). We were able to express our doubts and distress in a safe environment that helped us process this enormous change in our child.
We found both of these therapists through a list of recommendations provided by the children's hospital adolescent gender clinic.
I cannot recommend the therapy and contacting professionals strongly enough, even if you know for certain that you want to delay hormone therapy. Taking your child seriously is a step of validation that is enormously valuable. I think making hormone therapy contingent on a year or two of therapy can feel supportive to your child and could help you balance their wants with your desire to be supportive while postponing hormones.
What we did learn about hormones is that you can--and some people do--decide that they want to stop taking them. Many of the changes are subtle and reversible. Learn as much as you can about the trade-offs of hormones from a reliable, science-based source. You will need to know to help your child. We were well educated by our gender clinic.
At 6'2" with broad shoulders, my daughter does not "pass" very well, so anything to help her be more comfortable in her body was important to us. This is ultimately why we chose hormones when we did. Your mileage will vary, as your child can probably "pass" far better than mine and might be much more comfortable with their current body. If that is the case, great for you both.
As for the name change, we did it before the school year started because she wanted to apply to colleges with her new name and have her high school diploma in her new name and have substitute teachers know her name preference on the roll call sheet. Follow your daughter's lead in naming, though I think postponing the legal change until later, perhaps diploma time, might be ideal, though in the meantime, you can agree on a nickname. I have to confess that losing the name I had given my child at birth was painful and for maybe six weeks, I could only call her "sweetie" or such non-name things. A transition carries a lot of grief for a parent but you have to suborn all of that to help your child. This is a good reason to get your own therapist.
Now, a year into it, we've struggled a little. Getting the College Board folks in some conservative part of the country to stop hanging up on me when I tried to update her name was hard and involved talking to a manager and threatening a lawsuit. I did a fair handful of "coming out" moments as I changed her name with the doctor, the dentist, the optometrist, etc. The school system (we live in a liberal area) was easy and helpful. The judge who signed the name change was beaming and said this was his favorite type of name change. Her dad had to update his company to update the health insurance (this was very difficult for him, but his company was very supportive). He also updated the car insurance. The Social Security office guy was totally cool. The driver's license person gave her no trouble. The bank was crabby, however. Overall, it went much better than we expected, but again, we live in a liberal city. Nevertheless, I got to feel that nasty sensation in the pit of my stomach as I explained the situation and waited for the other party to respond. It's just one of those bullets that you take for your child. Your child has bigger fish to fry at school and other social and day-to-day realms.
Socially, everything is more positive than we expected. Save one teacher who systematically misgendered her, school was fairly easy for her. Students and (most) teachers were supportive. When she celebrated her 2nd anniversary with her girlfriend (who stayed with her through transition), some of her teachers even posted congratulations and many hearts for them on social media. She has plenty of friends. Overall, she is happier than I have seen her since about 5th grade. She still benefits from therapy but my husband and I stopped after a few months because we had stared all of our worst fears and worst selves in the face and had come to terms with that.
This is likely not a phase. The social cost of becoming the other gender is huge. Your child must be feeling very, very strongly to even tentatively voice their feelings. It's great that you want to be supportive and your concrete acceptance makes a huge difference in your child's mental health. As I have said, I recommend therapy for both you and your child. I also recommend that you try to transition to your child's preferred name. (Like I said, it took me weeks.) This will go miles for your child's comfort level and happiness. If your child moves back to their birth name, that's easy and costs you nothing. If your child persists (which the dr.'s said something like 95 or 98 percent do), then you will already be comfortable with that. I recommend the legal name change at approximately the same time we did it, just because it's easier to become a legal adult and get that sorted. And your child will probably want their new name on their high school diploma as well.
It's hard and painful for a parent to watch their child go through this, both agonizing over their child joining one of the more misunderstood and discriminated-against sectors of society and for you grieving over the loss of the child you thought you had. That is a totally legit feeling, though one you may not want to discuss with your child until they're MUCH further along their journey, if ever.