At 5'9" 180 lbs you are not that overweight. You would probably be best served by maintaining your weight and replacing body fat with muscle. Just get on a good progressive weight training program and eat at maintenance level. I suggest getting the new edition of
Starting Strength: Basic Barbell Training.
If you don't have space at home to buy an Olympic barbell set, power rack, and bench, I would strongly suggest you reconsider going to the gym. They will have tons of equipment you could never afford to buy yourself. I'm aware of what you said in the OP and your second post, but seriously, just go. You will see all shapes and sizes there and you will by no means be anywhere near the most out of shape person there. Everyone is there to improve themselves and you will get respect for doing the same. You are only 20 years old, the sooner you tackle this psychological issue the better, and you
really want to do it without becoming dependent on medication, if possible.
As mentioned, you can go at odd hours when the gym is nearly empty, you can wear an ipod so people won't try to talk to you, whatever you need to do. At the very least visit some gyms and let the salespeople show you around so you know what to expect before you go out and buy some piece of equipment which will give you very limited results, and which you might wind up not enjoying and losing your motivation for.