If I could save up $5000 by graduation do you think it’d be smart/possible for me to get a loan on a house?

Nobody replaces a functioning AC just because they've saved up. 

-Stay at home and save for another 12-15 months-Use FHA financing to buy a first multifamily property (2-4 units)-Live in one unit while renting out the rest-Pay down mortgage for 1-3 years, refinance FHA status into a conventional loan-Apply for another FHA loan and repeat the same process in another propertyObviously it would be better to buy and find a property that can cash flow.

@Tim White sounds like you may have to get a higher efficiency washer and dryer and do the Laundry onsite to save time.

IKI'm 23 yo and I've been saving and saving and working/working to try and save enough to qualify for an FHA in order to purchase a home. 

Save every dime you get.

I recently moved graduated from college, started a decently well paying job, and moved back home to save money for the next year or so.

I'm currently saving nearly 50% of my monthly salary. 

I’ve saved up a little nest egg and I’d like to invest it.

Learn everything you can, save up some money, and then you can start thinking about heading out on your own.