Well, it depends. If you don't have that much experience with horses, then I would recommend you to lease a horse. If you have been riding for quite a while and know a lot about horses than you can own a horse. So technically lease first then buy. It's totally up to who you are and what you decide but this is what I would recommend.
Even though I do currently own a horse, I'd have to say that I wish I could be leasing instead right now because I'm going to college in a couple years and I'm pretty sure I'll have to sell my horse whereas if it were a lease, I would just end the lease. (I know some people do own horses in college, but personally it is not realistic for me)
Both have their ups and downs. I lease two and own one. Leasing gives me access to high quality horses that I would not be able to afford otherwise. But I enjoy owning as well because it gives me the freedom to change disciplines or training methods as I please, as well as making general decisions based on my own views and beliefs, rather than someone elses.