A road bike should be faster than a mtn bike, particularly on hills. However, as Greg LeMond said, "It doesn't get any easier, you just get faster." I can easily finish my one-way commute 10 minutes faster on road bike compared to a mtn bike.

Start by looking at "sport touring" models that are relatively light weight but have mounts for fenders and racks and clearance for larger tires. Some examples would be the Salsa Casseroll, Soma ES, Gunnar Sport. If considering a used bike, a lot of older Japanese bikes were sport touring models. Look for brands such as Lotus, Miyata, Shogun, Panasonic, Univega from the 1980s.