How do analyze the error bars of a Standard Error statistical test?


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I'd be really appreciative if anyone might be able to help me. I'm studying biology at a foundation level and have recently been introduced to statistical tests for the first time as means of analyzing biological data from experiments. Because I've no previous statistics experience, I am finding it a bit of a struggle to get my head around how the basic Standard Error test works. What I specifically would like to understand is how the results of a simple Standard Error test with 95% confidence (95% confidence limits) can be analyzed. I know that you calculate the equation of Standard Error by dividing the standard deviation of the data by the square root of the number of samples. I also know that to calculate Standard Error with 95% confidence I must subsequently multiply the product of the Standard Error equation by 1.96.
At my level of biology I am then required to draw a bar chart with a bar for each sample being compared. Each bar should show the sample mean and error bars either side of the mean that represent the Standard Error of that sample. I then need to see whether or not the Standard Errors/error bars of the samples overlap. How should I interpret my results/findings?
If I calculate the standard errors for 2 mean values and then multiply each answer by 1.96 to get a value for each that is of 95% confidence (95% confidence limits) and then draw a bar chart depicting error bars for each of the 2 means, how would I then analyze these error bars? If the error bars overlap, would I say 'there is a 95% probability that the two means are not statistically different and that any differences between the two means are due to chance. Therefore the probability that the means are different is 5%?' If the error bars don't overlap, would I say 'there is a 95% probability that the means are statistically different and any similarities between the two means are due to chance with a percentage of 5%?'
Or would I perhaps express these ideas with different wording - for example by saying: 'if the error bars overlap there is a greater than 0.05 probability that any differences between the 2 means are due to chance' and 'if the error bars do not over lap there is a greater than 0.05 probability that any similarities between the means are due to chance' ?
Am I correct or incorrect with either of these?

Please might anyone be able to tell me if I'm on the right track?
Thank you very much - I'd be hugely grateful for any help!
I'm no stats expert, but I would only label your bars with 95% CIs, they're much easier to interpret than SEM imo. Also, it is absolutely NOT true that if CIs overlap, statistical significance has not been achieved (this is a very common misconception). Someone please correct me if I'm wrong, but CIs can overlap by as much as 25%, yet you can still have significance with p<0.05.

Also, be aware--statisical significance alone to a biologist doesn't mean much. You must also look at your effect size. For example "how much did X change in response to drug Y?" is a question a biologist is really interested in. A biologist isn't likely to be interested in a statistically significant result that is biologically irrelevant because of how trivially small it is. Statistical significance alone says nothing about your effect size, the reproducibility of your data, or if your hypothesis is correct.
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