First of all, you need your nexus file with the data matrix open, and with the tree / trees on which you want to trace characters imported.
To import a tree go to: "Taxa&Trees" menu, "Make new tree block from..." sub-menu, and click on "use trees from separate NEXUS file".
Your tree will be imported to a tree block that will be displayed in the left column: click on its name, and select "view trees".
When you have the tab with the tree, go to "Analysis tree" menu (the menu will not be displayed if you are on another tab) and click on "Trace character history", then ok in the dialogue boxes until mesquite displays a small box with two arrows, the name of the first character and its states. You will notice that the branches of the tree are shaded in different colours, to code the evolution of the character.
On this little window, you'll be able to select the character you want to see, and the colour of the branches will change accordingly.
Be warned, there are sometimes several possible optimisations of character states evolution, Mesquite will give you only display one possibility (I think it is Acctran, but I am not sure at all).
I suggest you list all steps in Paup to know which ones are ambiguous and when they are ambiguous, see which of Acctran, Deltran, Minf or something else make the more sense.
In regards with this thread, is there a menu in Mesquite equivalent to "apomorphy list" in PAUP which shows what apomorphy characters define a particular node in a tree? I have tried "State changes on branches" and a couple of numbers which seem to represent character state changes (e.g. 0-->1, 0-->2, etc) and their fractions do appear in a separate tab. But I can't tell what are the characters and associated states as they all appear as numbers.
The reason I am doing this in mesquite because I was trying to save the apomorphy list from PAUP (where I built the consensus trees) in text file but couldn't see the list in notepad and word document. It is also because I find out that the "trace characters" and "trace character history" in mesquite are useful per se.
I had the same problem, and ended up using Mesquite for tracing the characters history, but PAUP for a list of apomorphies. Unfortunately there does not seem to be an option for that in Mesquite.
After having analyzed the data in any program save the resulting tree in nexus format, then take the character matrix and (if you're working in mesquite) export it as a simplified nexus.
Then open both in PAUP with "execute [datamatrix filename;]" and then "gettrees file=[tree filename];". once these are in PAUP go to the File menu and select "Log Output To File" (this will print the apolist to a text file that is readable in Notepad). Then use the command "describetrees 1/apolist=yes". After that you will have a complete apomorphy list in your log file.
(I assume you know these command, because you used it before, but it took me a while to find it, so I'll leave it here for others).