I am really into Christianity and trying to learn and grow, but they still seem to be rather worldly. I am trying to lead by example, but I wonder if that is enough. The knowledge that they could die at any time makes me want to be rather hasty and get on them about things. Today I had a sort of argument with my mom. She says I am worrying about church things too much, and she didn't handle what I intended as constructive criticism. I tried to get across that what I was saying was for her well being and that I was looking out for her. But, anyway, things sort of got very emotional and didn't seem to end on a very good note. I am afraid I may have put some sort of stumbling block, if you will, in her path, like I didn't represent Christ very well. The emotions were more feelings of sadness than they were anger because I knew things were not all right. Was I being impatient? I was sort of blunt, but I did try and get across that I was trying to help. But more importantly what should I do now? I don't want to push her away from God.
The problem is that you are under your parents' authority and are at the same time trying to make them do what is right. Such normally doesn't work. You can't make anyone do what is right anyway, and your parents are confident that you know what you are talking about -- after all they raised you and not the other way around.
The reason that the Bible recommends setting a good example in such cases (I Peter 3:1-2) is because seeing the result of Christianity in your life will lead to them investigating and asking what made such a change in your life. Notice that in the way you approached this, you didn't show yourself different from the world. You let yourself get angry when people weren't doing things your way -- just like most people in the world.
If someone asks you for advice, then offer the best you can give. However, to insist on giving advice when none was desired is more often than not going to be rejected. A parent can give you advice because it is expected -- they are over you in authority and they have more experience. A child giving advice is not expected.
Instead, focus on being encouraging when they do the right things than rebuking them when they are wrong. If correction is required, then it has to come from a point of humility -- as soon as you start thinking or acting that you know better than they, you'll lose your point, even while being right. "And a servant of the Lord must not quarrel but be gentle to all, able to teach, patient, in humility correcting those who are in opposition, if God perhaps will grant them repentance, so that they may know the truth, and that they may come to their senses and escape the snare of the devil, having been taken captive by him to do his will" (II Timothy 2:24-26).