'My first word of advise is this: if you're a solo traveler coming to Moulins by train, pack light; the taxi's aren't that plentiful in Moulins, and it might just be easier to walk to your hotel. But don't worry - it's a small town and fairly simple to get around. The locals are very friendly and helpful.
I was there from the 20th until the 22nd of December, which coincides with their Christmas fair; they had all the holiday favorites including a carousel, vendors of heated wine, roasted chestnuts, and crepes as well as a troop of stilt walkers in beautiful fantasy costumes - one even played the violin.
The Centre National Du Costume De Scène is just across the Allier river - roughly 15 minutes on foot from the shopping district. If you're interested in stage costumes, this is well worth the trip.
As for restaurants, there's a place called Patapain that offers sandwiches, salads, and pizza-oid foods as well as various deserts and beverages. I recommend the hot chocolate and their flatbread pizza with salmon.; seriously, salmon on a pizza - give it a try.
If you'd like something a little more upscale, I recommend Pizzéria Valentino; I went to this little Italian restaurant my first night in Moulins, and it's a delight. It doesn't open until 7pm, but trust me - it's definitely worth the wait. The evening I was there, I had the Seafood with Rice... they didn't skimp on the portions or the flavor. And if you like organic foods and wines, that's another reason to visit Pizzèria Valentino.