Heavier karts have bad acceleration, so when using a heavy kart you have to get around that disadvantage by any means possible. That means getting the turbo start at the beginning, knowing how to power slide, and using the turbo pads on the track. The good thing about being heavier is that smaller karts won't knock you down or get in your way.
If you're able to keep from getting hit from other karts(not the items, just the karts), then you could switch to a lighter kart that has better acceleration.
Middle-sized karts don't have any real weakness, so they're good starting karts.
Also, since different characters have different special items, you could mix characters depending on their special items to get a good combo. Like the Toads' golden mushroom special can combine with a yoshi's homing egg special to provide you with speed and attack. If you have two characters with the same special item, you'd have less variety in special items but there'd be a greater chance of getting the special item between the two of them. Like if I had Mario and Luigi, it wouldn't matter who's in the back to get fireballs, while if you had Mario and Yoshi, Yoshi's egg would only appear while Yoshi's in the back. If you opt for double item blocks, they'll both get an item so you don't have to worry about who's in the back. Experiment with different characters to see what you work best with. Personally, I don't like Wario's bombs because you can get caught in the explosion, and I don't like Mario's fireballs because they don't aim well and don't travel a great distance. They can also bounce off walls and hit you.
This is all basic or obvious stuff, so if you don't know it, learn it. And if you do know it, practice it. Practice makes perfect, like they say.
