The problem is that ALL projected systems are made to...
- Give accurate measurements.
- Give a pretty picture.
An accurate measurement projection is for a small"ish" area. Not the entire world.
All this being said. Arcgis will allow (other software do this too) multiple projections into one or different projections.
So you just decide what projection "looks" the best for your map, Set the document projection (see previous answer from below) to that. and then load all your data.
The only reqirement is that each data source would need to have a projection defined. (shape files use a *.prj file, tif images use a *.tfw file).
These helper files tell the software what the coordinates relate to. On-the Fly projection does the rest of the work.
I suspect that because of the question you have one or more datsesets that are not overlaying correctly.
If this is true you should look at the prj file to determine if: first there is one; and second if it is defined as the correct projection.
You can search gis.stackexchange for other questions related. LIKE: "projections"
Or just look at the related to the right of this question.
If you look at the TOC (table of contents). You should have a data frame named layers (that is just a default name).
Right click on that and select properties.
Go to the coordinate systems tab and select the UTM zone you want.
Unless you are actually asking about how to show the utm grid (which is in the grid tab [same dialog])
this should do what you want.