Yes, definitely go clipless. You'll recruit muscles into your pedalling that have previously just been along for the ride. The laws of physics dictate that you still need to expend just as much energy to get over the hill but you'll be using more muscles to do so. As for the climb getting easier, the best way for that to happen is to start commuting every day. Within a few weeks it will be easier.

I use Shimano mountain biking shoes and SPD system, probably the cheapest model of shoe they make. The cleat is recessed into the bottom of the shoe and I find them perfectly fine for wearing all day.