When a rider must switch trains in route to their destination, can they get off and visit the linking town and then board a later train to their destination on the same ticket or must we buy multiple tickets in order to get off? Example: Oxford to Stratford upon Avon. We have to switch trains in Leamington Spa and would like to visit the village instead of simply switching trains. Does this require separate tickets for the journey?
That depends on the ticket/fare type purchased. Advance (discount) fares are sold for specific trains only, so you would have to purchase separate tickets. Off-Peak and Any Time (full) fares don't have that limitation, but may also be specific to one train company, not a competitor serving the same route. If there's a BritRail pass that fits your trip, then you're covered all day, regardless of which trains you take. Other Travel Cards also offer a more flexible way to get ticket discounts on short notice. http://www.ricksteves.com/travel-tips/transportation/trains/britain-rail-passes.
You can also combine Advance (restricted to one and only train) tickets for the long portion of the journey with an Off-Peak or Super Off-Peak (restricted to time and perhaps operator - such as only valid after 9:30 weekdays, anytime on weekends) for the short bit so there are no worries about times.
By the way, and it may have only been as an example - but having worked there for a number of years can I assure you that Royal Leamington Spa (to give its actual name) is no village. Not a lot for tourists to see in Royal Leamington Spa but the Gardens are quite nice.
I think that Warwick has more to see, including St Mary's Church, Warwick Castle, the river, the canal, the old medieval centre, and the antiques centres.