I don't know if you'd call me second generation or third generation Ukrainian. My father was certainly raised in a Ukrainian family, but he was born in Germany and grew up part of his life in Argentina. He also never taught me the language, so I can't help you with that.
Foods and holidays: Christmas Eve was pretty big in my father's tradition, and there was a special menu that went with it. I remember especially the Christmas borsht---it had ushki, or little mushroom dumplings, in it. Technically, it's supposed to be a meatless meal, which in the Uniate tradition means you get fish. (Uniates are sort-of-Catholic; basically, they're attached to Rome but have a special dispensation allowing married clergy and all the Russian Orthodox trappings that you might expect. My grandfather was a Uniate priest.) In practice, my mother had a fish allergy and passed it along to me and my sister, so we fudged on that bit. Dessert was medunek, a sort of honey cake, or kutya, which is raisins and nuts and things with honey. I have no idea if either of those are spelled right---they probably aren't---but spelling Ukrainian words in English is kind of an adventure anyway. Ukrainian uses the Cyrillic alphabet, just like Russian.
Oh, and speaking of Ukrainians versus Russians---it's not Baba Ludmilla, at least not in my grandparents' area of the Ukraine. It's pronounced "Babcha," and heaven only knows how it's spelled. Grandpa is "Dedo." You can understand Russian if you speak Ukrainian, and Ukrainian if you speak Russian, at least if the other person is willing to speak slowly and occasionally point at things, but they do not think of themselves as the same people. Russians call the Ukrainians "little Russians" sometimes. The Ukrainians are not amused.
Diminutives generally follow Russian rules, though, and a grandmother would probably use a diminutive in talking to her granddaughter---"Katherine" becomes "Katya," and so forth. And "Isabel" becomes "Izunya"---just in case anyone was wondering where I got my net-name from.
(No, that's not a traditional nickname, because Isabel isn't a traditional Ukrainian name. I think my father made it up on the fly. But it is a very Ukrainian-style nickname, and if your main character has some American-sounding name, I can see Babtsia Ludmilla doing something similar.)
By the way, the Ukraine hasn't had a very easy century, and depending on when your story is set, Babtsia Ludmilla might not just have emigrated, but escaped from the Ukraine. My Dedo (Volodymyr, in case you need a nice common Ukrainian male name) got his family out ahead of the Russian Army, which was transporting priests to Siberia, and ran into Germany---in the middle of World War II. Fortunately, it was the tail end of the war, or he might have escaped into something really ugly. You see, the Nazis had Slavs of all kinds catagorized as Untermenschen, but were willing to let them live so long as they took jobs that the Germans didn't want. For some Ukrainians, at least, one of those jobs was concentration camp guard. There were already really bad feelings between Jews and Ukrainians---the Polish lords generally used Jewish tax collectors when the Ukraine was under their rule---and WWII must have made it twice as nasty. When I was four or five, I remember a neighborhood woman deciding that I was an evil, unstable little brat and persuading her daughter Sarah to act pretty hatefully toward me. I didn't figure out until much later that it might have been because I was half Ukrainian and she was Jewish. So that's an ethnic vendetta you might want to know about. Whether it impacts your main character or not depends on how you want your story to go.
Hmm . . . can't think of anything else at the moment, and this post is pretty long already. If there's anything else you need to know, feel free to ask---with the caveat that I'm not very Ukrainian, so there's a good chance I won't know.
Izunya