Both stations are on the Victoria Line of the Underground railway system so no need to change, just take the Victoria Line southbound.from Euston.
If you have to drive then from Euston Station take ...
Euston Road,
Gower Street
High Holborn
Shaftsbury Avenue
Charing Cross Road
Trafalgar Square
The Mall ( Through St James Park and past Buckingham Palace)
Buckingham Gate
Buckingham Palace Road
Victoria Station.
That's 2.9 miles
About 30 minutes
50 minutes
You can't travel directly from Euston Square station to Victoria. If you go to Euston Station - a short walk - it's only four stops on the Victoria line.
Take the Victoria Line direct from Euston to Victoria. Alternatively, take a cab - however, it will be considerably more expensive and probably won't be quicker.
It would take about 1 hour to walk from Euston to Buckingham Palace.
The best option is to walk to Euston Square Station - which is well signposted and a two minute walk from Euston Station. Then, take the Hammersmith and City Line, Circle Line or Metropolitan Line to Liverpool Street.
About 45 minutes.
The Trainline web-site says - take a tube train from Euston to Paddington, then a first Great Western service to Reading. Journey time Euston to Paddington is 28 minutes and Paddington to Reading is 25 minutes.
About 4 miles and would take about 1 hour to walk.
Walk to Euston Square which is around two minutes away and well signposted and take the Hammersmith and City Line or Circle Line to Paddington.
Trains don't go from Kings Cross to Manchester. They leave from London Euston which is a 10 minute walk from Kings Cross. The journey from London Euston to Manchester Piccadilly takes around 2 hours 10 minutes.
Oxford St is 0.75 Miles