What are some good golf excercises I can do at my house to help me get better?

You can do jumping jacks, sit ups, push ups, and touch your toes with out bending your knees.

look up core training and do anything related to that..
most involve some type of strengthening the back, abs and turning around your hips, torso etc…

if you can get an elastic cord with tension such as lat pulldown
cord, etc.. stand sideways with tension and mimic your golf swing pulling with both arms and using your abs to pull the
cord thru your torso and back away from you, as far as you can, mimic your golf swing.. you can do it with a tension rope type thing you would stretch with that you could buy at wal mart.

take a medicine ball and rotate around your torso as many times a day as you can..find a friend and stand sideways and mimic your golf swing and play catch, if you use a brick way…

anything that makes your core muscles stronger or mimics your swing and makes you stronger and more flexible will
help trememendously.