I can't for the life of me remember what I used, but it was just characters that I had used often anyways, nothing special iirc.
Anyways, I prefer to have a medium kart. If you do have a light kart, be sure to have the Bullet one. Cheap as.
And all the cups are easier when you are playing 2P co-op. I find it so much easier concentrating on driving when my sister is able to throw the items so I don't get distracted by that.
However, a good tactic to learn is this. When there are two single item-boxes, and you can drive through both, hit one, hit Z, and then hit the other box. It'll be just like getting a double :D That could help you. Items are sometimes handier than you think...