I was working there for five weeks just before Christmas, so my experience will be a little different than a romantic getaway.
First its mega expensive, the £50 flight will probably be the cheapest thing you see all weekend. I liked were the old town, tram stop - place de neuve. There are some nice coffee shops and a few bars, and they sometimes have festivals in front of what looks like an old castle. My favourite place to visit was Carouge which you can get there on the number 12 and 18 tram. Loads of boutique shops, coffee shops and a decent bar called Bar Du Nord (doesn't open until 21:00 on a Saturday). There is also a market on a Saturday, which just before Christmas was beautiful.
Food is a tricky one, as there are loads of restaurants but cost a small fortune to eat there. If you are after somewhere cheaper the area around Plainpalais has a lot of different types of restaurant. Its near to the university and a bit sleazier than the rest of Geneva. I didn't sample it, but the smell of weed often drifted past.
Its busy with stuff around the main train station, Gare Cornavin. I didn't like it as it is much like other train station areas, and doesn't feel as safe as other parts.
If you stay in a hotel, public transport is free, and you should be able to pick up a pass from the hotel. Don't venture too far on a Sunday as the trams run, but the bus service is pants. The transport is top notch though, and the trams are great. You can pick up a free transport ticket from the airport to take you into Geneva.
If you are after culture the Red Cross museum is good, although not that big. The botanical gardens is also worth a look if that's your sort of thing. I guess taking a picture of the flags and big chair at the United Nations would be a tick box to do. I think there is a spa down near the lake that sounds good, although I didn't visit it. Probably the nearest you will get to swimming in the lake at this time of year.