I also want to learn self defense but I want to be a really good fighter in general.
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First piece of advice....don't feed the trolls
Next, there are a lot of people on here will a huge experience of all sorts of martial arts. However to give you proper information they need a bit more information. For example you say you want to become a fighter, what sort of fighter? Do you want to win competitions, aim for the Olympics, the UFC or just challenge yourself now and again?
Do you just fancy a stand up style or do you fancy grappling or both?
You cannot in reality do five styles and do all of them justice, your best bet at first is to try each of them and see what you like. See how you get on in class, see if the style 'fits' you and if it fits with your life. Then choose, karate and TKD are similar enough that you don't want to train both, BJJ and wrestling are also similar enough that you don't want to doth both at this point. Quite often people chose a stand up style and a ground style, just two as a beginner is enough, they won't confuse you and just doing two will give you the time to train properly. Years down the line when you know your core styles then branch out and have a look at similar things.
What you choose as your stand up and your ground style is up to you, as I said see what suits you, where you fill you fit in. Forget this 'street' fighting stuff, it's macho nonsense.
You need to think carefully about what you want then talk to coaches, the real experts who will help and guide you towards your gaols.