First, do serious research to see if there is another non-profit like your idea. You may be surprised to find out there is.
Then, read about your state's rules for incorporating a non-profit. First, you file with the state to become one, there is a filing fee. In state, the fee is $75. You can also register as a charity, in my state the fee is $15. You must provide your mission statement, make it general, in case your non-profit morphs into something else over time.
Eventually, you may want to file forms with the IRS to be tax-exempt. This process is lengthy & quite involved. You will need to seek the help of a professional.
As far as website design, check out godaddy.com for purchasing domain names. A website designer can help you with the basic set up so it is easy for all people to navigate.
If you know someone who already runs a non-profit, talk with them about it. One of my dearest friends runs a successful non-profit & set me straight about a lot of my ideas.
There are many steps involved and it takes a lot of time, planning and money to start-up and maintain. With dedication, your non-profit can be successful.