We regularly use trains to get out of London for our day rides. There is no need to book your bike on these trains on day rides (and indeed it is not normally possible).
As a general rule bike booking is for longer distance trains rather than commuter trains but you can get definitive information by checking the rail company websites.
Some mainline trains have a designated bike area but it is not normally necessary to always use it. There may be too many bikes in it already or it may be that you are joining the train at an intermediate stop and it isn't clear where to stand in order to get to the bike area in time when the train times.
Do use it if you can as it will make your journey more comfortable.
The main restriction for mainline trains is that you cannot take your bike (unless folded) on trains arriving in London in the morning weekday peak period or leaving London in the evening peak period. NB the start and end of the peak period is defined slightly differently for mainline trains compared with Overground, DLR, tube and Elizabeth line.
For the Overground, DLR, tube and Elizabeth line the peak period applies to trains in any direction. Outside peak times bicycles are allowed on the Overground, DLR, Elizabeth Line, the surface parts of the tube and the "cut and cover" underground lines but not in the underground sections of the other tube lines. Consult the relevant website if you are not sure as the tube is quite strict about this.