If I run a 10-3 with ground direct burial to a panel in a small garage and use it to split into to 30 amp curcuits what breaker do I need in my main electrical panel feeding the garage panel?

The breaker in the main protects the wire, so seeing as it is #10 you can use a 30A breaker.

What are you using 30A breakers for in the garage?

If this is for one 240 volt 30 amp circuit, why the need for a subpanel?

The breaker in the main protects the wire, so seeing as it is #10 you can use a 30A breaker.

What are you using 30A breakers for in the garage?

I'm feeding through to a panel in a chicken coup for lights an automatic door opener and outlets for heat lamps

Just know you cannot use 30A circuits and breakers for general 15 or 20 amp receptacles and lighting circuits.