It starts of in the key of E major, yep. At 0:40 you should be able to hear the key change? It changes to the dominant key: B major.
Stick to the E chord until 0:15. There's a bit that only has drums and one singer so it's hard to figure out the chord - perhaps go to an A or B major if you think they sound good.
At 0:15 (when the backing vocal comes in) it goes to A; when you hear the BVs go up a step, go to B.
The next chord change is a C sharp minor, followed by a D major. This is a "non-functional" chord as it isn't within the key of E, but don't worry about that nonsense.
Then for the "stick together" bit, you want a 1 4 6 5 pattern.
Every two beats, change the chord. The pattern is: E, A, C sharp minor, B.
Play that through twice until you get to the "side by side" bit. For that bit do E, A, B, Csharp minor and stay on that chord until the key change.
The pattern for this bit (play twice) is: B, E, Gsharp minor, Fsharp.
Then: B E Fsharp A. (Again, A major isn't in the key of B so it might sound odd)
I guess you should end on a Fsharp followed by a B.
This would be much easier in the key of C, especially considering you'd only have to have a key change to G which has much easier chords than B.