What pots work the best on electric ranges to sautee, fry and simmer?
While a cook will tell you that good pans are always better, I think that they will not solve your problem. If anything, better pans conduct heat better, so if you keep cooking at the same temperatures as now, you'll burn more food. A good temperature control with bad pans is useful, good pans with bad temperature control don't bring much. 

The correct solution sequence is listed below. You can stop at any point, as long as it has solved your problem. But getting later points done before the earlier ones is unlikely to do you much good (although in the case of new stove, it is understandable if you skip the point because of the expense). 


Whenever in doubt, use a lower heat setting. It is almost always better (and the result tastes better too). There are very few cases where this principle doesn't work (deep frying, not enough browning on the outside) and in most of them, you can correct if you raise the temperature in the end, after the food is cooked through (exception: deep frying). It is almost impossible that your stove cannot cook rice through; I have seen recipes for "cooking" rice at room temperature (it gets soft after a day or two in water). It takes lots of time to cook rice, about 20-30 minutes after the water has started to simmer. You are probably turning the heat up earlier, resulting in the burnt rice. 
Learn the correct cooking temperatures for different kinds of food and food preparation methods. There are books for that. The classic one is McGee on food and cooking, but many others contain this information. Having such a book is much more important for cooking than any recipe book. 
Utilize a method to recognize the current temperature of your food/cooking vessel. Easiest way with best results: get a thermometer (three is better - infrared for pans, flexible-connected feeler for roasts, candy thermometer for everything else. But start with infrared, you can point it at cooking liquids too). The so-so way: learn the common tricks and use them. Water is easy, you can see how much it is boiling. Oil is more problematic, but the water droplet should work for meat and the white bread for deep frying. I don't know of a good collection of all "recognize temperature" tricks, you'll need to hunt them here and there. Least helpful solution: learn from experience, developing a feeling for how long it takes a certain amount of food in a certain pot on a certain setting. Takes lots of trial-and-error, and lots of burnt food. Once you have to cook outside of your "comfortable area", you have a big chance to get it wrong. 
Upgrade your stove. I know, this costs a lot. But based on your description of the rice situation, your stove could be one of those which gets 85°C at setting 3 and 220°C at setting 4. These are a real pain in the neck when cooking, and in this case, even a €50 portable induction single-hob is a real improvement. If your budget and space allows it, it is highly recommended to consider it before upgrading the pans. 
Finally, upgrade your pots and pans. I guess that your problems with the burnt food will have gone by now if you have done the other points, but if you can afford it and cooking is a hobby for you, it gets much more pleasant with better tools. You get the best value for the money from iron, and good quality nonstick (with the mineral mix, or ceramic all the way, over a thick bottom) is almost indispensable for some applications (e.g. aerated omelettes). But if you really want to know what makes a great pot/pan, here is a really great article on the topic. It is highly recommended read for anyone who is serious about cookware. It should answer the question in the title too. While the author doesn't specify resistive heat, what he writes is suited for everything except induction (actually, the facts he describes are useful for induction too, but you have to make your own conclusions because of the different heating method - for example, tin-clad copper is one of the best things for outer heat, but definitely a bad solution on induction).