By road you can leave Henley in four main directions - north to Oxford, west to Reading, south to London and east to Marlow (back route - also accessible via the London road). Roads around Henley are not generally fast (with the exception of the M4 and M40 motorways), and overtaking can be restricted. Watch out for the many speed cameras in the surrounding counties, especially as you go through villages. You will be responsible for paying any fines you incur during your stay. The town speed limit is 30 miles per hour, main roads generally 60 miles per hour and motorways 70 miles per hour. Overtaking should always be done on the right. StreetMap carries gazetteers of most British towns and cities.
Railways are now privatised, which means that a long train journey could involve a number of possible routes and fares. Online booking is common but confirm trips just before you travel, in case engineering work has hit the line. (or the ubiquitous leaves). Henley routes tend to go through Reading as a connection and transfer point, and since it is a small town, trains are not particularly frequent. At Reading you can make connections to London, Oxford/the north, Bristol/the west and so on. The train station is located behind the River and Rowing Museum slightly to the west of the town centre, a quick walk from the Bridge. Try Railtrack for fares and travel news and The Train Line for price quotes and booking.
Coach travel between towns and cities is dominated by National Express, with selected routes run by small companies. Most Henley routes tend to fall into the latter category, and if you want to go on a National Express route you will probably have to start from Reading. Tickets may be booked by phone or online in advance, and it is recommended for most journeys. Most coach routes are served once or twice a day, with the popular ones as frequent as hourly or better. Henley buses to closer locations such as Oxford, Reading, Marlow, Maidenhead and so on tend to be regular but variably frequent - visit the Tourist Information Office for the latest schedules.
If you want to fly to the major British cities, including Edinburgh, Newcastle, Manchester, Glasgow and Belfast, you will have to do so from London itself. Other short-hops will be available on selected carriers. Use Cheapflights to track down the best fares. You won't need a passport (unless going to Eire, eg Dublin) and check-in is usually much quicker than for an international flight.
There are a number of boat trips which can be taken from Henley, a central point on the Upper Thames reach. Most of these trips are day-returns or shorter, but you may be able to negotiate a period return to allow you to stay for a night in a nearby town.