All of us want to live life as well as we can and even the most seemingly despicable character will usually live by some kind of moral code. The thing is that however warped it may seem, most of us do what we do because we think it's the right thing to do at the time. Unfortunately the fact that there is no official 'right thing to do' or 'right way to be' means that it can be hard for us to know how to act for best.
But at the same time, 'doing the right thing' isn't really enough to ensure that we are making the best possible contribution to everything. As well as doing the right thing you see, you also need to do things right. You need to do the right thing... rightly... yes it all gets a bit confusing.
Basically 'doing the right thing' means acting in the way that you think is right at the time. However doing things right basically means 'doing the thing you are doing already, correctly'. In other words, doing the right thing refers to the decision of what you do, and doing things right refers to what you do within that context. Doing the right thing might be choosing to help your Mother to move some boxes, but doing things right in this case would then mean taking time and care and being careful not to break any of her things (rather than rushing it and causing damage).
Doing the Right Thing
Doing the right thing means doing the thing you know will have the best long term effects for the greatest number of people (generally). It means foregoing what is necessarily convenient, comfortable or practical for you, and instead focussing on the 'greater good' or on making others happy and comfortable.
As mentioned earlier, there is no official 'right thing' and this is a highly subjective concept. However you should have a personal code of conduct, and you will certainly most likely know what is not the right thing, and you can use this to guide your decision.
Doing Things Right
Now bear in mind that if something is worth doing, it's worth doing well. In many ways it's worse to offer help or to try and do the right thing if you aren't going to put the effort in and try to get it right. Make sure that once you've made the decision to do something, you give it your full attention and you work hard on getting it right rather than doing a rush job. If you can't commit to doing things right, then you probably aren't doing the right things. See what I did there?