voltage regulators "speed" can cause dimming as wellif you do either, do the alt first.
think about it... your average battery puts out 12.8v TOPS... if your lights are dimming that means that your electrical system is sagging to your battery voltage. So a new battery doesn't fix the problem of your alt not being able to keep up, and you'll still get dimming (typically).
The alt on the other hand puts out somewhere between 13.8 and 14.4. if it can keep up with demand your battery never comes into the picture.