I have a cutler Hammer 125 amp main lug panel that I have installed for a sub panel to feed an out door kitchen and a sun room. The sub panel has 2 lugs for connecting hot wires and 1 lug for connecting a neutral. The sub panel will feed 4 ceiling fans, 9 lights,
1 bath heater, 1 refrigerator and 10 receptacles. The sub panel does not have a ground bus bar. Do I connect the ground wires with the neutrals? Do I run a ground from the 200 amp main breaker panel to the sub panel? If so, where does it connect since there is no ground lug? It is approx. 80 feet from the main breaker panel to the sub panel. What size wire would I need to run? What size breaker do I need to put in the main panel to feed the sub panel?

The largest breaker you'll be able to get will proably be 100 Amps.
If you were using 100 A, then #1/3 NM-B copper with ground can be used for 80'.

Can't size the panel because you left the loads out. This is a maximum.
You didn't give any loads and voltages of the loads.

You have to buy a ground bar kit for the sub-panel. Grounds and neutrals must be separate in the sub-pane. If the sub is in a detached structure, then a ground rod is needed.

4 wires need to be run back to the main panel. If copper, then 3 #1's and 1 #8.

Grounds and neutrals must be isolated in the sub-panel.