First of all, “Any Permitted” printed on a ticket does not mean that any conceivable route is permitted; instead it means that you can travel by any route which is a permitted route.

What is a permitted route, you may ask?

The National Rail Conditions of Travel states:

13.1. Your Ticket may show that it is valid only on certain train services, such as
those of a particular Train Company, or on trains travelling via a certain route or
routes. If no specific route or Train Company is shown, then (subject to any time
restrictions for the type of fare you have purchased) it will be valid on:

13.1.1. any direct train service between the station(s) shown on your
Ticket;

13.1.2. by any services (including any change of trains) over the
shortest route which can be used by scheduled passenger services
between the stations shown on your Ticket;

13.1.3. any other routes as shown in the ‘National Routeing Guide’

So your ticket is valid on the shortest route by rail used by scheduled passenger services, or on a direct train, or by a route permitted by the National Routeing Guide.

The Conditions go on to state that the easiest way to check the routeing guide is to check The National Rail Enquiries website. The idea here is that you type your ticket’s details into the journey planner, and set via point(s) to force travel via the route you want to check, and then see if it offers your original fare for sale or not. While this does work in many cases, it is sometimes hard or impossible to check more unusual routes in this manner, and the electronic routeing guide used by NRE is different to the human-readable routeing guide, and often forbids routes that the latter allows. Though do note that if NRE allows a route that the Routeing Guide forbids, then NRE takes precedent; this has also been said by the rail industry to apply to any properly licensed journey planner.

So if you really want to get the most accurate possible answer on whether or not you can go a certain route, you will need to learn how to interpret the National Routeing Guide. This is not straightforward but is definitely possible to learn. The basic procedure is as follows (though do please read the manuals before attempting this):

• For the origin and destination on your ticket, find out the suitable associated routeing points. There is now a tool to do this automatically on the Routeing Guide website, but it can still be done “by hand” using the Pink Pages and the NFM64 fares from September 1996, if you don’t trust the tool.
• If there isn’t a routeing point in common, then you can look up the mapped routes in the Yellow Pages. Routes will display one or more combinations of maps. You start at your origin routeing point on the first map, and trace a route along the maps without doubling back (passing through a station more than once), to the destination routeing point. You must use all the maps and use them in order. Doubling back IS allowed when passing through an intermediate Routeing Point Group, within the Routeing Point Group only (this is the group stations rule, intended to allow you to change trains at larger interchanges rather than being forced to change at suburban stations). To get from your origin station to your origin routeing point and from your destination routeing point to your destination station, you may travel on the shortest route by rail. If LONDON is a Mapped Route, this means you may look for routes from the origin Routeing Point to London, and then from London to the destination Routeing Point.
• You must then check that your mapped route does not violate any of the 32 pages’ worth of Easements, some of which are Negative Easements which disallow routes that would otherwise be allowed. Positive Easements of course can allow routes which would otherwise be disallowed so you may look here to see if there are any additional routes you could use.

As you can see, a bit of a convoluted process, but not an impossible one by any means (the checking for Easements is the only really painful bit, and in practice usually searching the file for relevant-sounding nearby stations will turn up any applicable Negative Easements).

So, let’s put this into practice!

Direct trains

This one is easy — there are no direct trains from Surbiton to Richmond so this is out.

Shortest route

The shortest route by rail is to travel to New Malden, then via Kingston to Richmond.

Routeing Guide

Routes within 3 miles of the shortest route

In addition to the shortest route allowed by the Conditions of Travel, the Routeing Guide also allows routes within 3 miles of the shortest route (in practice, National Rail Enquiries gives you even more grace on this). However, I don’t think there are any such routes. So, we will look for mapped routes.

Finding the appropriate routeing points

The tool tells us that Surbiton and Richmond are both Routeing Points in their own right, so this is straightforward.

Finding mapped routes

The Yellow Pages lists the following map combinations: EF+TW, and WV.

EF+TW allows us to travel from Surbiton to Clapham Junction on map EF, and then changing onto map TW, from Clapham Junction on a train direct to Richmond (but not one via Hounslow).

Map WV allows us to travel from Surbiton via Weybridge to Virginia Water, and then either direct to Richmond via Twickenham, or to Richmond via Hounslow and Twickenham (changing trains at Hounslow).

Travelling to Waterloo is NOT allowed by the Routeing Guide.

Easements

I have not found any relevant negative easements that restrict the above routes.

However, I have found the following positive easement:

000012 Journeys from Surbiton and stations west of Surbiton to Kingston upon Thames and stations north of Kingston Upon Thames, may double back between Wimbledon, Raynes Park and New Malden. This easement applies in both directions.

To me, this would imply that on our shortest route example changing at New Malden, we could instead travel on to Raynes Park or even Wimbledon, before doubling back via New Malden and Kingston.

So, the total score of permitted routes is:

• Surbiton – New Malden – Kingston – Richmond
• Surbiton – New Malden – Raynes Park – New Malden – Kingston – Richmond
• Surbiton – New Malden – Wimbledon – New Malden – Kingston – Richmond
• Surbiton – Earlsfield – Clapham Junction – Mortlake – Richmond
• Surbiton – Weybridge – Virginia Water – Twickenham – Richmond
• Surbiton – Weybridge – Virginia Water – Hounslow – Twickenham – Richmond

However, as mentioned in another answer, NRE will actually allow travel all the way via London Waterloo even though this is not a permitted route per the routeing guide. In my opinion this is definitely an unintentional anomaly, but while it exists you can use it since it’s a valid itinerary provided by National Rail Enquiries. Getting past the barrier staff at Waterloo might be a different question though.

(As a fun little aside, we can speculate on why NRE is returning the “incorrect” result in this case. The first observation to make is that maps EF and TW both go all the way to London, so were it not for the “no doubling back when tracing mapped routes” rule, this would be valid. So let’s start on the assumption that NRE for some reason doesn’t recognise the double-back as a problem and work from there. The Main Lines (to/from Surbiton) and Windsor Lines (to/from Richmond) sides of Clapham Junction station are actually encoded differently in timetable data — the TIPLOC code for the former is CLPHMJM and for the latter is CLPHMJW. In addition, Queenstown Road (Battersea) station is not encoded into schedules for trains on the Main Lines. For this reason, if NRE is incorrectly using the TIPLOC code rather than trying to convert to the NLC or CRS codes, it would see that there is no double-back between Vauxhall and Clapham Junction, as it would interpret the two sets of platforms at Clapham Junction as being different stations. But that explains Vauxhall — why would it let us go all the way to Waterloo? Simple, the Group Stations Rule mentioned above! Vauxhall is part of the London Routeing Group along with Waterloo, and so since doubling back within a Routeing Group en-route is fine, NRE will therefore let us go all the way to Waterloo.)

(this post assumes you are on a regular rail ticket that is not route or operator restricted, oyster/”contactless” has it’s own rules, which may be more or less restrictive than a regular rail ticket)

Can I use any route to get to my destination on national rail?

No, you can’t just take any route.

Unfortunately the rules on what routes you can and cannot take are a hellishly complex mess as the railway tries (and sometimes fails) to strike a balance between allowing people to take reasonable routes and forbidding people from taking unreasonable ones.

My advice for a regular traveler is to look your journey up on national rail enquiries or another site that sells UK rail tickets (national rail enquiries is the most official, but their site seems to often bug out if you try to specify more than one via point, also after adding a via point the page reloads, but it doesn’t actually update the train times until you explicitly tell it too) and see what it says about tickets for your journey.

If you get a warning about “some journeys require multiple tickets” on a national rail enquiries search, then pop down the “other tickets” button for the journey you are interested in to see if that particular journey can be completed on a single ticket.

For example, If I was at Surbition station and I wanted to get to Richmond Station could I take a train to waterloo and then to Richmond

That seems to be ok.

or take a train via weybridge and then go to Richmond

When I did a search with “via weybridge” it seemed to come up with journeys via weybridge and virginia water, which it thought were ok on one ticket, but also a journey via woking and clapham junction which it thought was not ok, Presumablly because of the rather large doubleback.

In the case of major disruption the usual rules are sometimes relaxed. You would have to check train operator websites or ask staff to be sure.

As a rule railway companies do not allow you to overshoot your station and return to it, although you often do not have to pay extra if you find out that your train did not stop where you expected it and you need to return to that station. They may charge you with an ‘traveling without ticket’.
(Added after reading the other answers:) It turns out that in the UK you may overshoot and return more often than in most countries, but within limits.

There might be exceptions to this rule, where overshooting and returning is so much faster or easier that they allow people to travel that way. In that case you should either find a printed (or online) source and have that handy when you travel that way or ask a train official about your intended travel and make sure you are allowed to do what you want to do.

And also, train companies in the UK often print on the ticket which train companies and/or which routes you are allowed to use.

When you get an ‘all permitted routes’ or some sentence like that, you must use a logical route which is as direct as you can make it. So no overshooting and returning.