The Historic Dockyard is the main reason tourists come to Chatham.  Once the nation's leading naval base, the Dockyard closed in 1983/4 and much of the area is being redeveloped as housing, business and leisure.  It's probably the best example of a Dockyard covering Tudor to Georgian periods for the most part, and contains a small number of period vessels.  To my mind, the best aspect is the peace and quiet, and wandering around the site soaking up the atmosphere - but for those needing more activity, there are numerous interactive elements, and holiday-time organised events.

Whilst in Chatham Maritime, why not look in on the Dockside Outlet Centre.  The selection of shops is not complete, but there's enough to justify a short visit, and the Downunder Café is to be recommended for refreshments.

Between the Dockyard and the Town is Fort Amherst, one of the chain of 'Napoleonic' defenses.  As well as regular opening, it's known for its events.

From the Dockyard/Maritime area, take the Medway Tunnel under the river, and head for Upnor (Right at the first roundabout, then all the way around the next one so you're coming back on yourself. Take the turn almost immediately on the left, and follow the signs).  Upnor Castle is owned by English Heritage, so if you're a member, you'll get in for free (or join once there!)  The town of Upper Upnor is between the car park and the Castle, and is pleasantly atmospheric.  The same can't be said for Lower Upnor, sadly.

There are some pleasant walks, both sides of the river, further downstream from Chatham/Upnor if you have time.  You'd be best to buy the relevant Ordnance Survey orange map to check out footpaths, or Medway Council do some leaflets, available from their tourist centres.

Don't forget Rochester while you're in Medway!