Are there any better ways of vectorizing photos than Illustrator's Trace? I don't have one of them Wacom boards so I can't really do this by hand.
You don't have to have a tablet to create paths with the pen tool. Thousands of users never touch a tablet, but still use Illustrator.
The best method for any vector art is to draw it yourself. It does take practice, but that is the best method by far. People spend years learning how to create vector art properly. It's simply not something any automated tracing tool can do with a click of a button. In most cases, tracing provides adequate results at best. In many cases, and for many user, "adequate" is just that for them and they are happy. If you want more refined artwork with better control over anything, you really have to create the paths yourself.
If you are using any automated tracing tool, you need to familiarize yourself with how it works. It may help to tweak values or sharpness of an image in a raster editing app, such as Photoshop, before trying to trace it. You may need to draw with pen rather than pencil if you are scanning artwork. You may need to adjust settings in the tracing tool to get the exact trace you are looking for.
There is no "magic" button anywhere which will make any tracing tool work for everyone in every circumstance. There are additional tools such as vectormagic.com. However, as with any tool and stated earlier, even that may not yield the result you want if you are unfamiliar with it.