Yes you need a ticket Auchinleck to Euston. Canning Town is another 40 minutes or so away by underground, but you don't need to worry about that now.
The route is Auchinleck - Carlisle, Carlisle - Euston.
The £121.90 fare is the flexible off-peak return. You can book non-flexible Advance tickets instead.
It might be easier to buy a return to Carlisle on the day, for £19.70, and then just book Carlisle - Euston, as this will give a better choice of fares.
Most trains are currently £41.50 from Carlisle to Euston, but the 14:49 is only £32.
Coming back the next day it's a similar story if not more expensive.
Most likely you are looking at £41.50 * 2 + £19.70 for the Auchinleck - Euston tickets, so around £102.70, which is a small saving on the £121.90 you mention, but not really great considering you are tied to specific trains.
That's from nationalrail.co.uk
Looking at nationalrailgamestravel.co.uk you can sometimes get slightly cheaper fares, but in this case not really - they are still around £103.
To Canning Town you can use your games Travelcard
