-you can walk, trot, and canter with total confidence
-you can circle, sepintine, and do figure eights at every gait
-you have an ok seat and can work without stirrups
-you are in complete control and can ride semi-difficult horses

I don’t care if you buy a beginner’s horse you need to learn to ride. An instructor will teach you confidence, control, care, and posture. You NEED to master that before you can buy your own horse. The instructor you mentioned doesn’t sound good. But you NEED to find an instructor. I ride twice a week and that’s perfect for my busy life but I get plenty of riding in. "basic rider" is pretty broad, If you can work for ride you’ll get a lot more out of your lessons. Try once or twice a week. That’s the best for beginners-intermediates then you can get your own horse or lease or ride almost every day. It’s not by time rather your intermediate it’s by riding skills, control, and confidence. It depends on the instructor, rider, and horses.
References :

Horses are very quick to pick up on a rider’s weaknesses and fears and will take full advantage of the situation – like, for example the horse refusing to go forward or where you want to, ie not to the gate. He probably was not frightened of the instructor and certainly did not want to buck you off as you say. He was just out to get his own way. You have a long way to go to being even a basic rider and I would not advise buying a horse and learning on that. It would be dangerous and you would end up being unhappy and more importantly, so would the horse. An unhappy horse is a sick horse and he will get frustrated and bored and sore from being kicked, poked, prodded and having no clear instructions. If I was a horse seller I would never sell to the likes of a beginner.
References :
Riding instructor. Bought a pony at 10 years old after 4 years of lessons. He nearly killed me on several occassions because I was not strong enough to deal with his issues. Spent more time in hospital than out when I had him!

Also, it’s obvious you’re a beginner because no intermediate or advanced rider would say any of that. You contradicted yourself when you said you were scared on the horse but then you don’t need lessons. Umm…..
References :

That lesson horse you were riding was testing you! You needed to make him move forward and give him a reason to respect you for his leader. He totally won with that little threat of maybe bucking. Most horses that are broke are just faking and they won’t even try hard to actually buck. I’m not saying that’s every situation or horse but it is true for most lazy horses.

If two lessons are all you’ve had then you are not ready to learn on your own. It’s after you can handle and ride a horse at the walk, trot and canter. When you can slow them down and stop them well. When you have a secure seat and feel confident with your riding abilities. And when you can ride different horses, taking charge of them and having them respect your aids. This is roughly what an intermediate rider is. You will know when you are at intermediate level, you will have a good feeling about your capabilities with horses. Another big part of being experienced with horses is how to read horse behavior. You will be much more prepared for alot of situations when you know what to expect. Look for natural horsemanship books, or any kind of training book. Parelli, Clinton Anderson, Chris Cox, John Lyons.
A weekly lesson sounds good, if you want to get more serious later on go ahead and take two, if you like. And everyone progresses at a different rate, but you should be above basic in maybe, ( I’m guessing ) six months to a year. But don’t worry about what level you are, enjoy your time with horses and soak up all the knowledge you can about them and riding.
Good luck and have fun!!!
References :
6 year horse owner, rider for 7 years.

The beginner says: My horse was uncooperative. He would not listen to me. He wouldn’t walk when I asked. He would walk back to the gate even though I didn’t want him to. He’s must be a bad horse.

The intermediate riders says: My horse was somewhat difficult, and my skills were not always up to the challenge. When he was stubborn, I had to work hard to earn his respect. When he would not walk, I first made sure that I was not doing something wrong, and then I learned how to use a crop to make him listen to my leg. When he tried to go to the gate, I learned how to use my seat and legs as well as my reins to take total control of my horse’s body so I could steer him wherever I want.

The advanced rider says: Some horses are challenging for me, but those are the horses who teach me the most and I welcome the opportunity to learn from them. When I feel my horse thinking about wanting to buck, I know how to put him to work so that his mind and body are engaged. Not only does my horse walk when I ask, but I can ask him to go from walk to canter and back to walk without a step of trot in between. I can pick up true canter or counter canter just from using my seat. My aids are so subtle and precise that that are nearly invisible. I control both my horse’s forward motion and his lateral motion. I can send him sideways with a shift of my weight, or keep him straight as an arrow. When I have trouble getting my horse to do what I ask, I know how to use various exercises to fix the problem and reinforce my aids. If I cannot seem to make my aids understood to the horse, I know to seek help from my instructor. And I know from experience that 99% of the time, the problem lies with me, not the horse.

So go out and get your perfect pushbutton schoolmaster horse, and life will be good for a while. But the truth is that every time you ride a horse, you are training it, for better or for worse. So, pretty soon your perfectly trained horse will start developing bad habits, which it will have learned from you due to your own inexperience. That said, I totally agree that you are no worse off bumbling around on your own than you would be paying for an incompetent instructor. So either find an instructor who is competent, or plan to ruin a lot of nice horses before you finally figure out how to ride (and I hope you have a fat wallet if you plan to go that route).
References :
B.S. in Equestrian Studies, Professional Barn Manager & Instructor.