Assuming this is your first time in Worthing and you have arrived by public transport, we'll start at West Worthing station.
On the south side of West Worthing level crossing is the Downview. Unless you are really thirsty the best bet is to head north and a short walk (about 5-10 minutes depending how fast you go) up towards Tarring. At the second set of pedestrian crossings just before a mini roundabout, turn left up Tarring High Street.
A few minutes walk up here and you'll find the George & Dragon. Usually a good selection of real ales (six) and a pleasant garden. Food is served lunchtime. Continue out of the G&D (as it is known) and carry on up the High Street (note the old Parsonage - an excellent restaurant), the maltings and closed pub.
The Vine is your next stop. It is only 100 metres from the G&D and has recently been done up. If you've been before note that the doors have moved! The inside has a bit more wood than it used to but the 9 (nine) real ales (including occasionally a mild - King & Barnes Mild in a pin). Are always on top form. Food is served lunches and evenings and Monday there is usually a band on about 8/8.30. The garden is great for families (children are not allowed inside.
Depending on your plans you can mosey between these two or head about another half mile north to the Thomas A'Becket (Gales) from here you can get buses back to town as well as Horsham and Petworth (the Numbers 23 and 1 respectively) or you can drift back to West Worthing and your train home.
"Whatever you do enjoy it!"