Report the result of the one-way ANOVA (e.g., "There were no statistically significant differences between group means as determined by one-way ANOVA (F(2,27) = 1.397, p = .15)"). Not achieving a statistically significant result does not mean you should not report group means +/- SD also. However, running post-hoc tests is not warranted and should not be carried out.
Firstly, you need to report your results as highlighted in the "How do I report the results?" section above. You then need to follow-up the one-way ANOVA by running post-hoc tests.
You need to perform the same procedures as in the above three sections, but add into your results section that this assumption was violated and you needed to run a Welch F test.
Recall from earlier that the ANOVA test tells you whether you have an overall difference between your groups, but it does not tell you which specific groups differed - post-hoc tests do. Because post-hoc tests are run to confirm where the differences occurred between groups, they should only be run when you have a shown an overall significant difference in group means (i.e., a significant one-way ANOVA result). Post-hoc tests attempt to control the experimentwise error rate (usually alpha = 0.05) in the same manner that the one-way ANOVA is used, instead of multiple t-tests. Post-hoc tests are termed a posteriori tests; that is, performed after the event (the event in this case being a study).
There are a great number of different post-hoc tests that you can use. However, you should only run one post-hoc test - do not run multiple post-hoc tests. For a one-way ANOVA, you will probably find that just one of four tests need to be considered. If your data meet the assumption of homogeneity of variances, either use the Tukey's honestly significant difference (HSD) or Scheffé post-hoc tests. Often, Tukey's HSD test is recommended by statisticians because it is not as conservative as the Scheffe test (which means that you are more likely to detect differences if they exist with Tukey's HSD test). Note that if you use SPSS, Tukey's HSD test is simply referred to as "Tukey" in the post-hoc multiple comparisons dialogue box). If your data did not meet the homogeneity of variances assumption, you should consider running either the Games Howell or Dunnett's C post-hoc test. The Games Howell test is generally recommended.
First off, it is not essential that you present your results in a graphical form. However, it can add a lot of clarity to your results. There are a few key points to producing a good graph. Firstly, you need to present error bars for each group mean. It is customary to use the standard deviation of each group, but standard error and confidence limits are also used in the literature. You should also make sure that the scale is appropriate for what you are measuring. These points and more are discussed in our guide on selecting an appropriate graph (guide here). Generally, if graphically presenting data from an ANOVA, we recommend using a bar chart with standard deviation bars.
Now that you understand the one-way ANOVA, go to our guide on how to run the test in SPSS here.