Aside from having to be at least 16 and not yet a homeowner, there's nothing stopping you from opening a Help to Buy ISA, unless you've opened a cash ISA this tax year already. The process of opening one is the same as with any savings account; you can open the account by contacting the provider you want through any of the 'account opening' methods the ISA allows, and they'll tell you what to do from there. To make the most of it, you'll want to set £1,200 aside to place into the account straight away.
If you see a Help to Buy ISA that offers a better rate, it's possible to transfer your current funds over to it. Contact the provider you want to move to so they can set things in motion through the proper channels. Do not take the money out of your ISA and then put it into a new one yourself, as this would count as opening a new ISA and could even have tax implications.
There is one exception to the rule that says you can't open and contribute to a cash ISA and Help to Buy ISA in the same tax year, and that's when a provider offers a linked ISA range. However, the interest rates on these combined accounts can be lower than those on a regular Help to Buy account.
Furthermore, as you're able to transfer your cash out of a Help to Buy ISA at any time (if the specific account allows unlimited access) and put it into a cash ISA or non-ISA savings account (you'll just lose the Government bonus), there's no reason why you shouldn't just compare Help to Buy ISAs and try saving in the best one you can find.
Note that you will only be able to open a Help to Buy ISA until 30 November 2019, which is when the scheme closes. After this, you'll still be able to save into such an ISA and claim the Government bonus until 1 December 2030.