What is the relationship between Bonferroni and Tukey HSD, are they just two methods of conducting corrections for post-hoc tests? If this is the case, could I say "post-hoc testing was conducted using Tukey-HSD adjustment/correction"?
Or is it the case that Tukey HSD is a procedure of post-hoc testing and Bonferroni is a correction method? If this is the case, could they combine with each other?
The main difference is that the Bonferroni correction can be applied to any set of independent $p$-values whereas the Tukey HSD method only applies to pair wise comparisons between means. Both of them control the family-wise error rate. The Wikipedia article on that subject compares them and offers some advice about ones which you have not considered. Note that if you decide to use Bonferroni there is a better method with the same assumptions due to Holm. This is also sometimes known as the Holm-Bonferroni procedure.
