TGD, you said the magic word... cartridges. I know EXACTLY what you're doing wrong and how to cure it. The cure however begins and ends with prevention and it's all in your hands, quite literally. Grasshopper, we can teach you.
First, razor burn is by getting a too close shave. What happens is that you remove too many layers of skin cells exposing the underlying living nerve ends. That's what razor burn is. It lasts for a few hours until the nerve ends die back.
OK, how it's happening... You come to traditional wet blade shaving from cartridges. That was the key to me knowing what's going on. Cartridges must have pressure in order to work. They require it and won't work without it. Blade shaving requires just the opposite. Trust me on this. You need to use almost no pressure during the shave. Safety razors work best without pressure. Look in the mirror and pay attention to your skin. If it's indenting under the razor even the slightest you're using too much pressure. You may need to lean way over to get close enough to see it. We describe this as using only the weight of the razor, or using "no pressure" and we mean it. Fight what you did to make cartridges work. That's what's causing your razor burn. When you finally realize you're using pressure and can conquer that the bladed razor will work fine. Too many of the wrong sort of passes wil make it even worse. Back off and make one pass until you learn what you're doing.
When we write about technique that's one of the things that's included. Another critical part of technique is really good lather, and don't think you must shave to practice making lather. You can practice lather making in your hand. Then rinse it down the drain and make it again. IMO, it's just wrong to practice making lather to shave with. That lather should be the perfect sort and not practice lather. You want to make mistakes when you don't have a shave in the offing. You don't mention what soap. If it's good soap you shouldn't need to add anything for great lather. If it's substandard ask how to make the good stuff. Maybe your water is at fault. But we need to know more to help with that. Some soaps are known to be easy to lather and some are just junk even for those of us with experience.
But coming to blade shaving from cartridges I would be willing to bet all of Andrews pay checks for the next year that it's too much pressure at fault and the number of passes making it worse. :-)
True story, those of you who know me just move on, you've heard it before. I have abuddy who I converted to blade shaving. I got him a Merkur razor on line and he happily went home and began to shave. I'd ask him from time to time how it was going and he'd lie to me and tell me that it was going OK, but... and then he'd tell me of a problem and I'd tell him to use less pressure and when he thought he was using less pressure use half of that pressure. This went on for months with the same scenario playing out many times. Then one Sunday he called me and told me, "When you mean use no pressure that's what you really mean, don't you!?". "Yup", I told him. Then he proceeded to tell me how he decided to do what I'd been telling him for months as a last resort before chucking it all in*, and that he got the best shave in his life, how it was counterintuitive to use less pressure for a great shave, yada, yada.
*when in doubt do what you've been told for months. :-)
So I tell you now to use less pressure. Far less pressure. Then use half again less pressure, and continue to use half again less pressure every time you shave. You need to unlearn what cartridges told you to use and you need to send that signal to the hands. So when I first began this post telling you that you had the answer in your hands that's exactly what I meant.
Oh, and don't forget to learn how to make great lather. It's a critical skill for a truly great shave. Never use substandard lather; unless of course you want problems during the shave.
If you want to move into more aggressive razors, and I'm suggesting nothing as there is no "natural progression" into more aggressive razors, but if you do decide you need a more aggressive razor everything you learned will be required to use them. So develop perfect technique now and use any razor you want to in the future. Or not. It's totally up to you; it's your skin that's in the game. The nice thing is that perfect technique now will lead to great shaves whether you move to more aggressive razors or not.
Other than not knowing what soap you're using I didn't see any glaring problems with gear.
Brian. Lover of SE razors.