This probably goes against what you want and possibly everyone else's advice but if I were building a home gym (with a limited budget - that would be in my case anyway) I would concentrate on getting a good quality adjustable bench (with proper padding) and a range of fixed dumbbells from light to heavy (you can always add to these as funds allow). For equipment I'd get a chin up/dip station if possible, otherwise don't worry. If you can get a barbell with plates that's also good too.
With this set up you can do a ton of exercises that hit every bodypart, both upper and lower body.
You can get dumbbells with plates but it's a faff changing all the time, especially as you need to lift heavier weights for single arm dumbbell rows, for example. That's why I suggest forking out more for fixed. You should also get very light ones that you can use to warm up, especially for the rotator cuff area.
I would also look into bodyweight exercises and callisthenics as these are great and very taxing (and you don't need fancy equipment).