I did the name change first through the courts in NJ where I live. 2 1/2 month process but did it myself. There is a $200 court fee plus you have to publish in the paper twice. However it was cheaper than going through an attorney where they wanted to charge between $900 to $1500.
I did everything at one time (name and gender), it was just easier that way since I had already been full time for over 3 months.
NJ has a special DMV form that must be signed by a HRT doctor or licensed therapist with your "new" name on it for a gender change. With that and the court order, DMV was fairly painless. They had to go to their supervisor twice, but I was in and out in less than 30 min.
Social security was next for me (NJ doesn't use Social Security information on you DL record), I had my HRT letter and they told me that because I had a NJ license with Female on it, that would have been good enough to change my gender but they kept a copy just in case. Two weeks later got my new SS card, but they misspelled my new first name even though it was correct when I left. I went back, they fixed it and go a correct one a week later this time.
Unfortunately I was born in Tennessee, so yeah, I'm stuck with male on it. Even worse in TN is that for the name change, they cross out the old one and type the new name on top and you can still read the old name.
I've already done insurance (changing gender dropped my rate a little for Auto), banks, credit cards, memberships, etc)
I still need to do passport, but first I need to pick up some more certified copies of my court order from the county courthouse.