Moving to Eastleigh (near Winchester)

Good location.

Have a look at the Brewdog Chaingang on Strava - there seem to be rides pinging off everywhere. Can't vouch as to their quality.

The Broleur community has some pretty active members on the south coast - they may be able to give good advice.

Sotonia have great kit if you like red and white

Eastleigh is a great place to move to - you've got the whole of the Itchen and Test Valleys on your doorstep and that means hills. South Downs to the north, New Forest to the west. Hop on the train at Eastleigh down to Brockenhurst and you're right in the centre of the New Forest with great on and off-road routes in all directions. Stay on the train and get the ferry from Lymington and you've got the very cycle-friendly Isle of Wight to play on. 

There's purpose-built trails at Moors Valley near Ringwood. If you're into longer rides the area west of Fordingbridge (20 miles from Eastleigh) is brilliant - rolling terrain, lots of very quiet back roads.

Southampton claim to be working hard on improving cycle access and to be fair there's a fairly safe and mostly shared-use route out west from the City Centre to Totton, which gives you access to the New Forest from the South East.

Bike-wise if you want to take full advantage of the New Forest you're probably looking at discs and 28mm tyres. Most of the routes across the forest are flat hard-packed gravel, think enduro or bike-packing if you come from the off-road world or gravel/adventure if you're a roadie.