You could ride the Castleman Trailway into Poole. It's between an hour and 15 an hour and 45 mins ride depending on the exact route, (about 15 miles from Ringwood to Poole). It's mostly flat, fast fire road type riding which could easily be done on a road bike with chunky tyres.
Having got there, have an ice cream, wonder around the quay and appreciate the motorised 2 wheelers if you go on a Tuesday evening. Then turn around and go back.
If you are out for a short ride then Moors Valley Country Park are putting in some nice singletrack at the moment and it is really close to Ringwood, you can either go in the main entrance or come up the back alley as it were off the Castleman Trailway (They have some quite cool technical bits in one area including some half finished 'North Shore' and a pump track, or there are just miles and miles of forest roads with intersecting singletrack. But still very flat.
If you want long and fast with some good lung burning then the above will provide you with the opportunity, but as I have said it is all flat.
Loads and loads of riding if you head out over to the purbecks, although for some reason I haven't actually done more than a 2 hr ride myself yet! But there is loads to do here and the riding is apparently epic! Although still one for the short travel bikes to keep it interesting.
If your that kinda a guy the local club Dorset Rough Riders do a lot of rides in the local vicinity – check out the calendar for details.
If you go further afield to west Dorset/east Devon border there are some proper epic distance rides you can do, but there is between an hour and an hour half's driving to get you there from Ringwood.
This book by Colin Dennis is pretty good, I have it and have done a few of the rides, but I would very much advise sitting down with an appropriate OS Map or GPS and working through the route first, some of it is a bit confusing if you miss a junction (I found it hard to use whilst cycling so would strongly suggest using a GPS and setting his co-ords as waypoints.)
I hope this helps.
Dom