・I am glad that things around your family are going well.
> I am glad that your family is doing well.
Note: I would not use the word things, unless you are going to define what things are.
・It must be a great vacation to see your family and spend in California.
> That vacation you took in California must have been great. You were able to spend quality time with your family.
・Are you going to get to California by car or plane?
> What travel method do you plan on taking to visit California?
Note: I would ask method, because they might not take a car or plane. Maybe they ride a motorcycle or take a train.
・I can start getting paid when I get my permit to work.
Note: This sentence does work. However, people usually would say: I can start getting paid after I receive my work permit.
・I want to believe that I do not make things too hard.
> I would like to believe that I do not make things too hard.