The clicking sound on the outside unit is coming from inside somewhere. Would a bad contactor cause this or is there something more going on?

I had just replaced the transformer. After replacing that is when the clicking started, until it suddenly stopped and now appears the new transformer broke as I don't have 24v anymore.

Jumper your hot thermostat wire to the (probably yellow) stat wire,that is sending power to the a/c,if the a/c works then you`ll know its OK,and the tstat is probably bad ,if a/c still not working check for low voltage power at the contacter ,possible torched by the replacement of the transformer... you also can jumper the hot wire to the white leg and see if it operates the furnace,, also make sure there are no breakes or shortages in the tstat wire,, sometime the dog or weedeater make for a short life on tstat wires going into the condenser unit

There is no low voltage power at the thermostat or contactor or directly from the new transformer. Something caused it to burn out again.

Would a short in the contactor cause clicking sounds at the thermostat and from somewhere inside the outside unit?

A shorted out contacter will blow you transformer at times I`ve seen it happen,, the tstat might click a few times before the transformer blows again,, if you not careful you going to blow the tstat also,, the next transformer you install put a inline in so the fuse will blow and not the transformer,, but you going to soon spend enough in transformers and tstats that you could have paid a tech to come out and fix your problems