Below are my suggestions. Other people might think of alternative options. You might find
this map useful. Although it is not the most up-to-date version, it is more useful in your case as both Epsom and Bookham are included (Grid M6). The most up-to-date version is
here (which only includes stations within Zones 1-6).
You could buy the following for each day of your stay:
Epsom - London Zones 1-6
Route Any Permitted
Off-Peak Day Travelcard
£7.85 Adult £2.25 Child
(Valid on trains arriving at the London terminus after 10am, so basically anything after 9.20am from Epsom is fine.)
plus
Boundary Zone 6 - Bookham
Route Any Permitted
Off-Peak Day Return
£3.35 Adult £1 Child
(Valid on any train after 9.30am. This ticket must be bought from a manned ticket office.)
The first ticket allows you to make a return journey between Epsom and the boundary of London Fare Zone 6, plus unlimited travel within London Zones 1-6 for the whole day subject to time restrictions.
The second ticket allows you to make a return journey between the boundary of Zone 6 and Bookham. If you wish to visit Bookham first, then go into London on any particular day, just use the second ticket first (and effectively start "short" from Epsom). Between the boundary of Zone 6 and Bookham, you can either go via Effingham Junction and Surbiton, or via Epsom and either of the two Ewell stations.
(There is also a
Route Southern Only version of the Day Travelcard which costs £6.55 Adult and £1.90 Child. This ticket is not valid between Epsom and Ewell West, only Ewell East. If you get the wrong train you might be liable for a Penalty Fare of £20 per person. To keep it simple, I recommend just getting the
Route Any Permitted ticket mentioned above.)
All fares quoted include Railcard discount. I have simplified the time restrictions for you. Some exceptions apply but if you stick to the times I quoted above, you should be fine. Time restrictions only apply on weekdays.