Current suspension on my 12 Wrangler JKU is pretty stiff and I would really like it to ride like my 04 Suburban or 91 Caprice Wagon. Not sure it's reasonable to expect that out of a lifted Wrangler, but I'd like to get as soft as possible
Lose the lift kit and put the original suspension parts back on it. Every lift kit I've experienced stiffens up the suspension considerably from stock, even though some are advertised as "soft ride."
Definitely do as @doc said to make an improvement, but even stock it's still going to be a stiffly-sprung vehicle with solid axles front and rear. Not a recipe for a cushy ride.
Not sure it's reasonable to expect that out of a lifted Wrangler
To be honest youre lucky if its stable. Short wheel base, high center of mass. Handling on those is not good.
The suburban and caprice feel much better because they have a longer wheel base, and are probably considerable heavier too. Either remove the lift or live with it.
Depends on your budget. With suspension parts, you get what you pay for. There are a lot of good jeep shops out there; tell them what you want and how much you're willing to pay and see what they say. Just be careful; I don't think you want soft and sloppy, you want smooth and stable.
Agree with returning to stock height, if not an inch or 2 lower. And don't forget about wheels and tires. Off-road tires are pretty heavy and tend to rise rough (and unstable on pavement). Lighter weight wheels and high quality all-season tires (eg, Michelin Defender LTX; Continental TerrainContact HT) will improve your ride even if you don't change the suspension (though would look weird if you still have the lift).