When ate you coming? I'd do a pie, play and pint at lunchtime, then elk down Byres road , stopping in at the Huntarian museum (great shop). Make your way to the city centre, for cocktails and dinner, and then go to a gig in a pub or a ceilidh or something.
The Sharmanka Kinetic Theatre Sharmanka is well worth a visit and definitely fits your criteria of quirky and out of the ordinary! It was set up by a Russian artist/sculptor who emigrated to Glasgow in the late 80's/early 90's and is fascinating. It's in the Trongate area which is only a few minutes walk from one of the main shopping streets in the city centre and next to the Merchant City which is a fab area for a wander round. There are a few galleries around there too (including Street Level Photoworks) and great cafes and restaurants, including a Russian one, Cafe Cossachok around the corner from the Sharmanka. (I'm not Russian, just happen to love those places!).
The Gallery of Modern Art is free and right in the centre of the city. Sloans has a Friday night ceilidh, I've never been but have heard it is good, it is also apparently the oldest bar/restaurant in Glasgow.
Ooh these are all great suggestions. Arriving 18-21st Jan. Have just booked tickets to see The Maids at a theatre. What is the best (cheapest?) place to go for food? Was in Brick Lane, London a few months ago and loved it. Something lunchy but good quality/value.. informal. Quite fancy Indian but very open to suggestions
The Wee Curry Shop on Ashton Lane (2 mins walk from Hillhead Subway station) is Indian and good. Closer to the city centre is Trans Europe cafe which is good value and has a great atmosphere - old bus seats for chairs and maps as wallpaper!
If you like Indian food, I really think you can't beat the Mother India Cafe just opposite the Kelvingrove museum. Indian Tapas! It's brilliant but doesn't take bookings, the queue can be out the door but it's worth the wait! We also like Firebird, just down the road from there.
Museums are good (and free!) - Burrell Collection in Pollok Park, Kelvingrove, People's Palace, Tenement House, Museum of Religious Life and Art, Provand's Lordship (oldest house in glasgow) and Scotland Street School. Centre For Contemporary Arts, the Tramway and The Lighthouse can have good exhibitions, and Glasgow Film Theatre sometimes shows some really quirky or old-fashioned films. If the weather is good the Botanic Gardens and Kibble Palace are nice.
There are some good pubs especially in the West End, several often have ceilidh music (some with Gaelic names can be good). I'd avoid the city centre at night on a weekend unless you're into clubbing and stuff - it's not unsafe as Glasgow is a friendly place but it's just full of youngsters getting wasted especially on Sauchiehall Street!
Hope you have a great time, we love being weegies!!