"is the needle rotated correctly?"
This was the largest headache in figuring out my hand-me-down machine, a older Kenmore.
I had re-threaded the machine multiple times, trying different size needles, and finally sitting down and reading the manual front to back. I had missed the step where the needle had to face a certain direction. This fixed all my issues in not sewing through thicker fabrics, thread tension, and thread breakage.
It sounds like it is your bottom tension that could be off though. Is the bobbin wound properly?