Hi,
There are a few shops opposite the entrance to the Bel, which sell souvenirs, water, snacks and cigarettes etc. Although for proper shopping you would need to go into Hammamet. This is walkable (about half an hour I've been told), or it's a cheap taxi ride - about 2TND to the centre of Hammamet.
I've never done the quad biking myself, but have seen groups of quad bikers out in Hammamet. If you ask at the hotel they should be able to advise you. For excursions, I enjoyed the Carthage /Sidi Bou Said /Bardo Museum trip (and I don't usually much like museums!). The Garden of Hammamet trip involves spending most of the day on the bus, so wouldn't recommend it unless there's something on the route you really, really want to see. At the Bel you will see trips advertised, run by TWS. This is owned by the same people who own the hotel group. Ask around the other guests to see if they would recommend them - unless they've got their act together more since I used them (years ago now) I wouldn't. If not check with your rep what they can offer, or look into diy.
The beach is indeed right on the doorstep, and if you can't find a spot on the Bel's beach, you can walk along to the Royal or Sol's section.
There is nightly entertainment, but don't expect too much! Having said that, as you've been to Tunisia before you pobably have a fairly good idea of what's on offer! If things haven't changed, there's nightly bingo, followed by a show. I found the best shows to be the snake charmer and the fakir show. Most entertainment is hotel based, and while you can go to other hotels for their entertainment, in general it's pretty much the same everywhere. The only hotel I can think of where it seemed a bit more polished on occasion, with a slighly different programme was the Fourati (of the ones I visited). I seem to remember they had a foreign dance group in sometimes - but this will depend very much on the night. The Fourati is on the other side of Hammamet to the Bel so you'd need a taxi to go there. They have sometimes been a bit funny about letting non guests in, but I think this applies more to locals (I'm often taken for Tunisian!!
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Hammamet can be lively (if you're not expecting Magalluf type lively!) and it can be fairly dead. Outside of August it's pot luck (though personally I find August too crowded). As for bars, they seem to come and go and move around regularly. My favourite is Havana (again you'll need a taxi). It's billed as a club but is more a late night bar, with the club El Pacha next door (same owners). The bar doesn't start filling up till quite late though.
The Bel does have aircon, and as far as I remember it was fine. It did break down for a couple of days, in the entertainment room, but they just moved the entertainment outside. I'm sure there are internet cafés in Hammamet, but am not sure where. Internet access is definitely available at the Royal (stayed there last year)- I think it was 5TND for an hour. If you can find somewhere outside the hotel it will be cheaper.
Enjoy your hols!
A.