I'm a novice without a doubt, but not enough of a novice to go cross grain.
Planing against the grain is not the same as cross grain. Think of a wad of straws at an angle you want to plane in the direction of the rise, or else your blade will get "hung" on the edges and you'll get tear out. This principle applies to both edge planing and face planing, although in face planing it can be much more of a problem. I don't know if you're doing it, but starting out edge planing is much easier.
I think the long and the short of it is that the plane isn't sharp enough / configured properly.
You're definitely on the right track there. As mentioned, getting the back of the iron perfectly flat is step one - which means you have to have the proper sharpening stuff to do it. It also means you have to have decent stones and develop some sharpening/honing skills.
My question was about whether you can get a cheap plane to "work", not about whether you can get it to work as well as the much more expensive counterparts.
Yes, you can get them to "work", but is that really the right question? I firmly believe our skills will develop according to the quality of tools we use. I've learned this through experience getting frustrated like you have not knowing "is it me or the tool". Then only to find out later I DO have the skills but a crummy tool was getting in the way.
Bottom line: Use inferior tools, you will get inferior results, plain and simple.
This is why I really discourage someone starting with cheap tools like this. I'm not saying you need to run out and spend $350 on a plane, but would you spend $150? If so, I fervently recommend the Wood River #4. I believe it is the best buy on the market today. You will have a tool that will definitely "work" with no fiddling or second guessing.
I did a lot of research around this, and I actually found a video of a guy endeavoring to see if he could get the plane to work well just as an experiment. He goes through all of the steps necessary to clean up factory defects and get things working in a more predictable fashion. In his case, he wants to use it as a scrub plane, but even still. I've been pulled in other directions and haven't been able to try it, but his results gave me enough information to explore the subject myself.
A scrub plane is a whole different animal so you can't apply what he's doing to a regular bench plane. A scrub is a roughing plane not a smoothing plane, so you don't need a quality tool for this.
I would encourage you to check out several sources on tuning up and sharpening a plane because you need these skills no matter what you do. The guy who mentioned using a file - DON'T DO THAT!!
Someone also mentioned Chris Schwarz. He has an excellent series on tuning up a plane.
Good luck and welcome to the world of hand tool frustrations!!