You can put bar gel underneath the handlebar tape to make it feel pretty plush. You can also wear padded gloves. YOu should also move your hands around while riding - Don't keep a death grip on the handlebars - you can ride up on the hoods, you can hold the handlebars next to the stem, whatever, just move your hands about. As you build up your core, you won't lean forward so much and put weight on your hands. Keep your elbows slightly bent while riding.

I don't know if you've had the bike fitted to you or not - but it's possible that raising the height of your handlebars will make your position a little bit more upright and get some weight off your hands.