I think this doesn't work as well with the last bit, but maybe it's
Potential Spoilers ahead.
I am the very best
Like no one ever was
No one else (that we know of yet) can match him (Saitama). I don't recall for sure, but I don't think there's been an enemy yet that lasts more than a single punch soo...
To kill them is no real test
He kills monsters/aliens in one punch.
To find someone is my cause
I will travel across the land
Searching far and wide
A couple of the episodes are of him going places in search of a worthy opponent.
Teach my disciple to understand
The power that's inside
The cyborg hero kid (Genos), but he follows Saitama trying to learn and develop a similar strength.
It's mostly just me
I dont care about my destiny
Saitama is just doing things mostly because he's bored.
Oh, I have some friends
In a world we must defend
He groups together with some other characters to defend from, iirc, an alien ship or something.
A heart so true
My laziness will pull us through
He's bored mostly, but also isn't super motivated (?). Also, he's pretty blunt/forward and honest about how he feels and what his understanding is.
You teach me and I'll teach you
Saitama also learns some things from Genos, such as learning about the hero association. (@Jannis)
Every challenge along the way
With hope I will face
Saitama is hoping to find something challenging again.
I will battle every day
To claim my rightful peace
In the episode with the house of evolution his peace is disturbed by the attack. (@Jannis)
Come with me, the time is right
There's no better team
Arm in arm we'll win the fight
It's always been our dream
Perhaps this refers to that final fight of season 1 with the monster ship?
Yeah, apparently my memory is a bit fuzzier on the subject than I thought. Just waiting for the next season before I rewatch.
Some of these are still a little rough. Thanks to @Jannis though for helping me fill in the gaps.