Beginning Zephyr
Before you use Zephyr, you will need to have your data matrix ready to go - aligned, and with characters excluded should you wish. The matrix should be in your Mesquite file.
If you wish to run the phylogeny inference program on your computer, you will also need to have GARLI, RAxML, PAUP*, or TNT installed, and you will need to know in what directory it resides. If you wish to use the CIPRes service, you will need an account on CIPRes's REST service system. For PAUP*, this will need to be the command-line version of the program (not the version within windows and drop-down menus, etc.). You may need to know some details about the version of the program (for example, for RAxML you will need to know if it is the PTHREADS version or some other version).
With this in place, choose an item from the Analysis>Tree Inference menu. You will then be presented with a dialog box to choose the source of the matrix to be analyzed:

The typical choice would be to choose a matrix that is stored in your file, i.e., choose Stored Matrix.
You then might be asked a question about specifying an outgroup (if you have taxon sets defined in your file). But more typically you will be presented with the main dialog box for setting options for the inference program. For example, here is the dialog box you will see if you are running a RAxML analysis on your local computer:

There are tabs at the top of the dialog box ("RAxML Program Details", "Replicates", "Character Models & Constraints", and more if you touch on the triangle pointing to the right) that allow you to set various options. Once you have set them, and press OK, the RAxML search will start.
If the external program is GARLI, RAxML, or PAUP*, and the analysis is a search for the optimal tree for your data (as opposed to a bootstrap analysis) then as the analysis runs the current tree will be shown in a tree window within Mesquite. This is not available with bootstrap analyses.
If, after the analysis has started but before it has completed, you save your Mesquite file, then Zephyr will save into the file details about the ongoing search sufficient to reconnect with it once your reopen that file.
Once the analysis is done, Mesquite will harvest the resulting trees and present them to you in the tree window. If you have done a bootstrap analysis, then Mesquite will show you a majority-rules consensus tree of the bootstrap trees:
