You're going to get a whole flood of responses on this, buckle up. I think mine may be a little different than everyone else's or I wouldn't bother posting it. At the end of the day, I'm just some guy on the internet with an opinion, like everyone else, you'll have to make up your own mind.
First, I think the overly-bantered concepts of "on the street" and "protect my family" are almost throw-away concepts in the martial arts community. Your streets are likely different than my streets. You are likely different than me. When martial artists, especially this generation of martial artists talk about "the real world" it seems to be a notion of after-school fights. Two or more people agree to meet someplace and duke it out. This was reality for me 40 years ago, but it looks nothing like my reality (or likely yours) now. Think about what you are really concerned about as you go about trying to answer that question for yourself. When you hear or read someone say "on the street" or "for real" ask yourself what they mean and how closely it matches your reality. Better yet, get really in tune with your answer and stop listening to everyone else who thinks they know better.
"Protecting your family", (something I personally relate to) is something else entirely. This is much more complex and involves a whole bunch of variables that I don't even want to start trying to outline here. But, it goes way beyond punching and kicking and grappling.
So, next I want to ask, why did you chose TKD? If you're questioning your choice and looking to other arts to make up for perceived deficiencies after one lesson, I think you may have chosen impulsively. I spent a year doing TKD in the 80s, which doesn't make me an expert on it, but if you are only 1 lesson in and you're planning on mastering not only it, but also a supplemental system to achieve your goals, then I think you're setting yourself up to fail. If you were years into it, I might give you different advice, but I'm never in favor of beginners trying to cobble together something better than what they are studying.
It is necessary to buy into what you are studying for it to have a chance of working. Maybe TKD would support your goal, I don't know nearly enough about you and your goals to say, but if you doubt it, you either need to put that doubt aside, put your head down and train or you need to stop and assess alternatives.
If you were coming to me inquiring about my system, though, I would be asking you leading questions about what you are really trying to prepare yourself for and my advice to you would be to first get clear on that, then understand your options, then make a choice and don't second guess it for a few years while you train enough to reassess.