You want to prune it in the late fall/winter when its somewhat dormant or fully dormant.
I'm not sure the rule, but you don't want to cut further than 12 inches from the main branches.
Are you looking to reshape it or stop it from growing upwards?

I wanted to 'clean' it up.. It has approx. 10-12 branches coming from the ground. and they are w/in a 18" dia. I thought it would be healthier/happier, if I did a little nip/tuck... But maybe not...

I'd like to keep my fig tree from taking over the back of the house, but I'd also like to try rooting some of the cuttings.

The trick is finding a place to keep the bagged cuttings warm in a cool house -- would a heat mat for seeds work? And any advice on keeping the cuttings from growing mold?

Rhet

Here is a link that might be useful: What the figs from this tree look like

Our 4 year old fig tree is about 10 ft high and 10 ft wide. It think it would benefit from pruning and besides that, it shades my greenhouse now. Is fall or spring the best time to prune in Texas near Dallas? How do we do that so that we can root the cuttings? It made a two big crops of figs so I don't want to mess up the production with poor pruning.

Try removing four or so of them and see what happens next summer.

If you live in an area where the tree is susceptable to being killed to the ground each year, the recommendation is to prune it to an open shape with 4-5 main trunks. The trunks you keep should be at least 4" apart. Also, when you prune the top, make sure the top-most leaf bud is facing away from the center of the tree so the tree continues growing outward. Good luck!

The best time to prune depends on whether you have a Common-type or a San Pedro-type fig tree. Common-types should be pruned when dormant. San Pedro-types should be pruned right after harvest that way they will produce new branches in the current year which will bear fruit next Spring. If you wait until dormant/Spring, you will be pruning away your fruiting buds for the breba crop. Good luck!