Deciding whether to include all the covariates together, or looking at them individually (or n small groups), depends on your hypotheses. If you include more variables the meaning of your coefficients change. For example, if you just have Age as a predictor then the coefficient is a measure of the unconditional association between Age and the response variable. If you add say gender then it is the association conditional on gender.  Can you give more details of the hhypotheses you wish to examine? 

Thanks for replying, I wish to measure impact of demographics (independent var.) on the awareness level (dependent var.) of rural people about formal financial services.

By "impact" do you mean measuring the influence (i.e., cause) on awareness? If so, it is worth seeing Holland (1986, JASA, http://people.umass.edu/~stanek/pdffiles/causal-holland.pdf) on what can be a cause. Pearl (2009, Pearl et al, 2016, is more introductory, his web site is http://bayes.cs.ucla.edu/jp_home.html) discusses what can be done when you are not doing an experiment. 

Dear Jagdish,  

Firstly, conducting an analysis with all of the independent variables or just a few of them will almost certainly give you a different result (unless the effect related to all of your independent variables is  non significant, which would indicate that your sample is homogenous, or your selected independent variables have little impact jointly or separately on the outcomes).  

If you want to work out the importance and the interactions between the dependent and independent variables  you may find the best way is to apply a Univariate General Linear Model (UGLM).   This allows you to look at the effect for all of the independent variables together in a single analysis.   What you can then do is to remove non-significant variables from the calculation until you are just left with the significant variables.   In practical terms this is quite straightforward if you use a tool like SPSS, because you identify the least significant variable and remove it from the list in the next iteration of the calculation.   Keep doing this until you have only significant variables left, or the dependent variable is the last one.

The nice thing about this approach is that it is quite quick, and it allows you to look at the interactions between the dependent and independent variables much more effectively.  

I hope that this information will be of help.

There are several reasons to avoid choosing which predictors and/or products of predictors by looking at there individual achieved significance levels. If you do not have a theory for how things should relate I recommend using methods like the lasso (http://web.stanford.edu/~hastie/StatLearnSparsity/). Bruce also talks about looking ``at the effect.''  The suggested statistical procedure do not mean that you will be measuring effects of one variable on another. For causation further assumptions are necessary. See the Pearl reference I provided above. 

Dear Daniel,

Thank you for your clarification in relation to the word 'effect'.   You are quite correct - I was not careful enough with my wording.   However, to put this into context I was looking at the term 'effect' in the wider sense, rather than its purely statistical meaning.

Dear Daniel,

Thanks for suggestions and wonderful references.

Dear Bruce,

Thanks, I will try UGLM procedures. Can I try interactions even if no literature is available regarding the interaction of variables.