Home gym is definitely cheapest, but you also have to assess cost in terms of the actual benefits you get. Most gym memberships are not used (only 10% still go after the first three months of their 12 month contract), and most gym equipment bought folds away for easy storage for a good reason.
So you have to have a serious look at yourself, are you motivated enough to work out on your own, or will joining a gym or hiring a coach/trainer be better for you. I mean, is it better to spend (say) $1,000 and fail, or spend $3,000 and succeed?
It could be good to find a mate or two and work with them, share equipment saving you money, and your mates can motivate you a bit. I remember Adrian (Hulk) was offering his home gym in Sydney to a training partner, I might do that myself soon down here in the southeastern suburbs.