I'm not coming to the absolute defense of Mathematica. And no, I don't work for Wolfram (or have any commercial connection). But I have used Mma. for several years, and I have written some large computation routines with Mathematica. I would agree that the software could still do with some improvements.
It appears to me that your biggest problem with Mma. is in learning and understanding their languge. It's certainly true that the language is not very intuitive. It takes some real effort to master the approach they are using, especially rule-based programming. However, if you can get past the learning curve, the language does work just fine.
Ff you're simply looking for a quick-and-dirty method to solve math problems, it's probably worth checking other symbolic software. There have already been some suggestions on this thread.
However, If you'd like to become more proficient with Mma - now that you've purchased it - I highly recommend the book : "Mathematica: A Practical Approach" by Nancy Blackman. Ms. Blackman has done a lot to "explain" the Wolfram system to the rest of the world.