I grew up near Winchester, and still have mates there and my brother lives in Eastleigh.
It’s a nice place, the town centre is quaint I suppose, and it has nice restaurants and pubs. The cathedral interesting for an hour or so, and there’s a Ghurkha museum (linky), and a load of other military museums (the Royal Green Jackets for example).
I grew up in a small town nearby – Whitchurch – there’s the silk mill there which is interesting for kids I suppose (but I haven’t been there since I was in the cubs about 25-30 years ago!).
You could also go to Watership Down for a walk but that’s about 20 miles away near Kingsclere. The author, Richard Adams, lived in Whitchurch and there’s a pub called the Watership Down near Whitchurch (in Freefolk on the Basingstoke road). It’s got a nice beer garden and used to do good food, although I haven’t eaten there for about 20 years.
Stonehenge is about an hour away too, near Salisbury.
Bring your bikes too, as you can jump on the South Downs Way from Winchester…most of the organised SDW events start in Winchester.