at the possible risk of stating the bloody obvious -
from City Airport, you want a DLR going to Stratford International, but get off at Stratford - not Stratford High Street and not Stratford International.
On the DLR, either buy a ticket from the ticket vending machine at City Airport station (these are usually before you start going up / down stairs to the platforms) or tap in (and remember to tap out when you get to Stratford) with an oyster card or contactless card. DLR generally just has card readers (something with a yellow circular pad) not barriers.
You will need to go to the national rail ticket office (or a national rail ticket vending machine) at Stratford to get your ticket to Cambridge, as the oyster / contactless thing only works in (and at a few stations just outside) London. You might save a few pence by doing this at Tottenham Hale not Stratford, but not sure it's worth the effort.
I haven't the foggiest where the ticket office is at Stratford - I haven't quite got the hang of Stratford station since they rebuilt it before the olympics.
Some longer distance trains strongly advise pre-booking, but it's not compulsory on more local trains like this. And the chances of the railways being able to restrict number of people getting on at intermediate stations like Stratford / Tottenham hale is unlikely.
Booking in advance might just save you a few pence (and time and effort in going to the ticket office at Stratford) but if you do, make sure it's not tied to a specific train, some advance tickets are valid on the next train if you happen to miss the one you were going for, some aren't.
This is Greater Anglia trains website - I've never tried buying tickets online from them so can't offer advice.
And you do need to wear a face covering of some sort (unless you are in one of the exempt categories) on the DLR and trains.
View attachment 225074
from TFL's journey planner (although not giving time to go and get ticket at Stratford) - I did have to reject a few offers of slightly faster but less convenient routes.)