It depends. I can only speak to how things work in the US, so ymmv.

Teaching High School: for the most part, you would need a teaching license issued by a valid license conferrer, typically a state board of education. Attaining one of these usually involves a college education, and these days you'll reliably need to have some kind of masters to keep a long-term position (in many places, if you don't already have one, your district will push you to work towards one). Many postgraduate programs in teacher training will require a BA in history or a related discipline as a condition of entry. Personally, I would not recommend you pursue a bachelors in education if you want to teach at the secondary level, as ed degrees aren't really useful for much else.

The thing to understand about teaching high school is that a lot of schools are not really looking for history teachers. They want someone who can teach history, but also do other things for the school. This is one of the reasons so many American HS history teachers are coaches; it is very common for schools to look for people with athletic coaching experience or, say, debate or Model UN experience when filling these positions. As a practical matter, public schools tend to get assessed based on reading scores, math scores, and how well the school serves EL and special education populations. History just isn't really that much of a priority. To be blunt about it, schools tend to be much more interested in how into working with kids you are than how into history you are. The landscape is slightly different if you teach private, as licenses are not necessarily required, but they operate on similar incentives and you'll make less money.

Teaching College: If you want a career job teaching history at the university level, you'll almost always need a doctoral degree. This is, suffice it to say, expensive and time-consuming. You also have to be able to navigate successfully the academic politics of whatever program you land in, which can be easy or hard depending on what field your in and what university you study at. You also have to understand that there may not be a lot of jobs teaching what you want to teach. There aren't that many career history prof positions in comparison to the number of candidates competing for them. It's an extremely competitive landscape. You probably already know this, but you should have some proficiency in a language other than English, as this can be necessary for research. Plus, you have to be prepared to take jobs in places you may not want to live or work in. Doing a series of one year visiting fellowships at a succession of different colleges and universities is not at all unheard of.

The questions you need to ask yourself are: why is it that you want to do this, and what are you willing to do in order to attain in. As one of my undergrad profs explained to me, if you want to be history prof because you really like history and like learning it, you don't actually get to do very much of that once you get into a position because your time just gets taken up with other things.

I'm not saying this is something you shouldn't do, just be aware that there are costs and it may not end up being what you think it is.