From my experience, pain in anything is bad, unless you are talking about the immediate burning sensation in the muscles being worked. Lower back pain is usually a sign of something wrong- either structurally, strength imbalances, or improper form. If tightening your core muscles makes it go away, you probably need to work on lower back and ab strength.
Heres an article about why form is so important.
If you workout in a gym, I'd suggest having an experienced trainer check your form and show you some lower back exercises.
There are loads of free weight lower body exercises. Take a look at exrx.net and click on some of the lower body muscles to see which exercises to do.
Here's a short list.
Dumbbell and single dumbbell squats, free bar squats
weighted lunges- the combinations are endless
deadlifts
romanian deadlifts
one legged dumbbell deadlifts
Mel