Strangely enough, I'm actually driving from Podgorica to Tirana tomorrow morning. I've been based in Podgorica for the last 2 years after 7 in Tirana, so it's a route I've done regularly over the last few years.
From Podgorica, easily the best route is through Hani i Hotit. From there to Shkodra is motorway all the way (Podgorica to Shkodra is 60km and easily easily doable in 75 minutes allowing 10 minutes for the border crossing). I've done fthrough Ulcinj too, but that route through the border is much more..........rustic shall we say.There are no trains that cross the border, but there are a few coaches each day from Podgorica through the border to Tirana.
Lake Shkodra is stunning, and you'll get great views of it on the drive down through Hani i Hotit. Shkodra itself is worth a pit-stop and a walk around, but if your itinerary allows, I'd recommend Tirana every time.
As for driving, it's perfectly safe in Montenegro, but once over the border the Albanian style is a bit more unruly with more two lane roads. Nothing to be fearful of though.