Sensible logic to focus on a city, albeit with good 'spur of the moment' options outside. I know Torino is well-served with these, but expect Udine would be too.
I've not been to Udine, but it was on my radar, as much for seeing the subtle change in culture for a city on the north-eastern fringe. Just as Piemonte / Piedmont retains some French influence, there will be other subtle influences for Udine.
As that's my experience, I'll focus on Torino. It is my favourite city in Italy with visits well into double figures. With each visit there are new finds.
It's well covered for rain / thunderstorms, with extensive porticoed streets. The public transport system extensive and their map is either the work of a genius or a lunatic, and I'm still uncertain which, but I enjoy poring over it to work out a good route (AI be damned, doing it yourself is a lovely little puzzle)
Art and museums covered well, and it should be quiet, albeit there's always a chance of hitting a school visit. When I first visited in mid 1990s, Torino was not a tourist city, a local being both honoured we'd chosen his city, yet bemused we had "It's like me going to Coventry". Tourism has slowly grown, but it's absolutely a functional city first, with tourism a modest sideline, though the local tourist office is good. In particular, the quirky is very well-served here, from producer visits, to the museum of the mountains, to museo Pietro Micca and the underground counter-mining tunnels, tourism here is wider than the obvious.
Food is very good indeed, and although Eataly burst out from its Lingotto start point to take on the world, it's never my focal point for food grazing. I consider it not a food temple, but rather a phenomenally good supermarket. Instead it's the classically Italian multi-generational artisan shops that provide great excitement. Some close, but others do open up to take on the responsibility that good food deserves. I'll typically pick up bits here and there for an ever-changing brunch/lunch/supper/easy meal at the apartment, plus some stuff to take home. The daily market at the north of the city (Piazza della Repubblica) can be a little variable, but there's plenty of quality in there, and I'd absolutely recommend it. Eataly serves as a final catch-all for anything I'd not found out and about. When eating out in the evening, the Torinese take their evening aperitivo seriously, and what was intended as a few appetisers to go with a drink, at times gets out of hand with lovely buffets of food, that can negate any desire to actually have an evening meal. It's around 6:30pm-8pm. One dish I'll specifically call out, amongst a few traditional stars, is Vitello tonnato. Lots of variety in how it's made/presented, but it's a surprising mix of thinly sliced veal, with an tuna, anchovy and caper sauce, that I love. Historic cafes are well worth trying.
Chocolate another speciality with well into double figures of exceptional makers. My favourite is Odilla on via Fratelli Carle (number 37-41 IIRC), which has an initial reserved formality, but they seem to quickly warm to visitors to the shop, and I've been offered many tasters there on my visits. I like their balance of tradition and creativity. In other sweets, Leone pastiglie are very traditional small hard sweets in many flavours (the subtly charming historic al Bicerin cafe has a huge selection), but I'm not sure they're for me. Instead, though hard to find (there's a sweet shop in the covered market portion of the Piazza della Repubblica market that sells them), are another traditional sweet called Ginevrine (or Ginevrone for a bigger ones). They're simply gently melted sugar with very subtle flavourings and subtle natural colouring. The sweets are formed by gently pouring them in the form of small discs on greaseproof paper (I've made them at home, and they're easy to make). Nice little instant energy.
Wines of the region are another love, and not just the prestige nebbiolo from the Langhe region, but also the likes of Freisa/Grignolino/Brachetto/Vespolina/Timorasso and the wines of Ghemme, Gattinara, etc. alongside others. Also some very good beers brewed here.
Oodles of history here, from the landmarks of reunification of Italy, to the bridge Napoleon built, which they thought to demolish, but justified its retention by virtue of them being able to walk all over his construction. If you ever watched the original Italian job movie, you'll recognise many locations, some humdrum, some prestigious. A couple of grand country piles are in easy reach for any fair weather days, and likewise Basilica di Superga, which is also houses a shrine to the all-conquering Torino football team whose plane crashed into the hill. They've never recovered that position of greatness, but remain the popular team in the city (rather than Juventus, who are more global).
The Torinese themselves? It's the north, so a little more formal than the south, but in my experience no less friendly or engaging. Being lightly touched by tourism, there's little to none of the exploitative aspect that can blight tourist hotspots.
Logistics? Aside from the amazing route map, the train network is good with local and fast freccia routes offering a wide variety of day trip options, albeit a little confusing with multiple hubs, seemingly competing against each other (Stazioni Porta Susa, Porta Nuova, Lingotto and the very odd stazione Dora whose station serves the airport, but is away from the centre - the shuttle bus typically a better option). The airport mid-sized, so has a relatively relaxed, yet not uber-efficient vibe. The in-airport 'experience' modernised for the winter olympics, but as ever with airports, not necessarily improved.
I hope this gives a taster, and no slight intended to Udine. I'm sure I would enjoy it.
