Haven't played Mario Kart since N64. Girlfriend and I have been looking for a game to play together since we beat Snipper Clips... I guess this is a no brainer?
I really, really like Mario Kart 8. The Wii U version retroactively destroyed all other Mario Kart games for me.
That said, I've found that Mario Kart actually isn't the best two player game. The track is kind of empty unless you add in CPU's, and then you'll spend a lot of time competing/interacting with the CPU's instead of each other. Going online together helps a bit, but you're still not interacting much with the person physically next to you.
I actually think Mario Kart is a lot better either online or just in single player. Four player works well enough but in Mario Kart 8 that cuts the frame rate in half, which freaking sucks.
Try playing the Grand Prix cups in multiplayer as a co-op experience. Here's how my friends and I play:
We make a a goal of one of us getting first overall in each cup and collecting the first-place trophies for every cup. (You might also make a goal of earning 1st place in every individual race and ending with a perfect score if you want to make it harder.) When you're first learning courses and playing on the harder difficulties (or just new to MK), getting first can be a challenge, so having a teammate can make it more fun and easier. This team approach introduces more strategy and complicates the game a bit. In our case, we'll actively work to make sure certain people keep finishing in first—for example, sacrificing a 1st place for your friend because you finished 6th on the last race but your friend finished in 2nd—to ensure a win at the end. Additionally, some roles start to develop, as people who aren't in first should be on the defensive, preventing CPUs from gaining on the leader and warning the leader when any offensive items are headed their way. Those people should also still want to finish as best as possible because it pushes all the CPUs lower on the point scale and because the best defense of a 1st place racer is an allied 2nd place. It really gives everyone more to think about because they're thinking about how they're doing individually (like normal) but also as a team.
My words might make this sound complicated and thus more work than fun, but as long as you think of yourselves as a team with a common goal, all of the strategy and tactics should come naturally and with little thought.
When my friends and I play this way, we are constantly communicating and have a blast. I think we developed this rhythm with Double Dash and have made Mario Kart a co-op experience ever since. I pretty much exclusively buy the MK games for the co-op experience they provide me and my friends.