Carollo's Italian Grocer & Deli in the City Market would be my first suggestion. I know there's also a booth at the actual city market which has fresh pastas every weekend, but the name escapes me.
For outstanding gelato... I'd actually recommend hitting up Christopher Elbow's new ice cream / sorbet / gelato outpost Glacé. It is phenomenal.
20 E 5th St, Kansas City, MO 64106