The Affordable Care Act makes great headway on that. Any plan available on the Marketplace, Medicaid, and Medicare can't discriminate against transgender people.
What that means - being transgender is no longer considered a pre-existing condition that would make it difficult or impossible for you to obtain health insurance. You can also not be charged more due to being transgender. This is a great start. Prior to the ACA, I had to be really selective about which of my healthcare expenses I ran through insurance to avoid tipping them off that I was trans. This ruled out having any kind of surgery and having my testosterone paid for, because there is really no medical reason why someone listed as female needs that and no way for a surgeon to advocate for a procedure as medically necessary without saying I'm transgender. My doctor had to code all of my visits as a "non-specified endocrine disorder" to get coverage for the visits and my lab work.
Now, in terms of covering those things now. It all varies state to state. Regardless of where you are, if you get a marketplace plan, you should be covered for the basics: mental health services, preventative care examinations, and hormone replacement therapy. Currently, a lot of states have provisions that would deny coverage for surgical benefits related to transition with exclusions like: "sex change operations", "transsexual surgery regardless of medical necessity", or "procedures performed in connection with sex reassignment", etc.
But - starting 1/1/17 - gender identity and sex stereotyping will be banned for most types of health insurance coverage. This makes those broad exclusions above invalid. People will probably still encounter issues as companies make the switch, but all complaints can go directly to the Office for Civil Rights. There are also quite a few advocacy groups that offer free assistance if you find yourself in that situation. Out2Enroll is a great one.
I think you'll find also on an individual level that a lot of doctors are willing to go to bat for us in gray situations. My current plan initially denied covering my hysterectomy bill because I'm listed as male on insurance. It was a big, whopping $40k bill. I didn't even get a chance to appeal it. When I went to see my doctor at my follow-up appointment, she told me not to worry about any bills I get from insurance - that my portion would not exceed my deductible for the year, and that she had already appealed the decision and contacted the office in charge of reviewing health insurance discrimination claims. True to her word, I got a bill about a month later that was for exactly my deductible. Shortly after that, I got a letter from my insurance saying they had reversed their decision and paid up.
So how do we continue working toward making healthcare accessible? Supporting the Affordable Care Act. Tell your representatives that you support it. Write testimonials about your healthcare experience and how it differs now and has impacted your quality of life. Ask your healthcare providers to write in support of it from a professional standpoint. Support insurance companies like Kaiser that by and large provide coverage for transgender care in every state. Read up on insurance in your state and help others locally find plans that will work for them.