The old (medieval) town sits on hill west of the river, the new town with the railway station is on the east bank.
From Nice it can be reached by the SNCF (the french railway company) train via Monaco. The adult fare from Nice is € 6.40 for adults / € 4.80 for youth (12-25 years) (as august 2010). For tickets you will often need the amount in coins or an EC/Maestro debit card. Beware that in Ventimiglia the ticket automats are often out of service; those traveling here for a day trip should purchase return tickets before they start.
The "new" town center and shoreline are an easy walk from the station toward the south and shore. The original/historic section is across the river to the west...also an easy walk.
Ventimiglia has an awesome medieval old town not designed for tourism, but rather an interesting place where normal inhabitants live. It has also a beautiful view of the beach in its highest point.
Elsewhere, you can see the remains of a Roman theatre (century 2nd ) and the traces of the ancient city walls. The ruins of the ancient Albintimilium are situated in the plain of Nervia, c. 5 km to the east of the modern town.
Go to swim at the beach!
The "new town" has a decent selection of storefronts offering quality goods at some savings. You'll find decent bakeries/pastry shops everywhere, but for better mercantile, you should venture at least a few blocks away from the train station.
The covered market opens every day except Sunday, offering seafood/meats, fresh vegetables, fruits and locally/regionally-made sauces, jams/jellies and such.
On Fridays, the "big market" comes alive. Stretched many blocks along the shoreline on Passeggiata Cavallotti in the new town, vendors hawk rather repetitive offerings of often counterfeit goods...clothes, purses, shoes, etc., at very "competitive" prices (after bargaining). Savvy shoppers occasionally find genuine, unbranded bargains...often made in Asia.
If visiting by car on a Friday, parking can be very difficult to find.
Travelers entering France by car on Fridays from the direction of Ventimiglia are frequently questioned by police at the border about counterfeits; much less often, westbound trains stopping here may receive similar screening. If found, any counterfeits that you have will be confiscated, and you'll likely be levied a major fine.
The train terminal contains an indoor buffet restaurant that enjoys a decent reputation for its food and cost. For more sophisticated offerings, try near the seashore in the "old town", e.g., Pasta & Basta.