He maketh me to lie down in green pastures: he leadeth me beside the still waters. (line 2)
The happiness of the speaker is innocent and pastoral – the kind of thing you see in a lot of nature poetry, both corny and not. "Pastoral," by the way, that means having to do with shepherding. A lot of more modern writers have seen shepherding as the ultimate quality-of-life profession – you get to spend your days in the outdoors tending to adorable little lambs.
He restoreth my soul (line 3)
In the third line, the speaker shifts from talking about physical happiness (rest! nourishment!) to spiritual happiness. Like the pastoral landscape, the speaker's soul feels fresh and new.
Yea, though I walk through the valley of the shadow of death, (line 4)
Well, this poem certainly took a leap into darkness. It's as if the shepherd took a wrong turn from green pastures and suddenly landed in death-haunted valleys. But the speaker makes clear that this dark part of the journey is unavoidable and may even be essential.
thou anointest my head with oil; my cup runneth over. (line 5)
The happiness is in the second part of the poem is social, and returns to the idea of physical nourishment and even luxury.
Surely goodness and mercy shall follow me all the days of my life: (line 6)
Perhaps the most important part of the speaker's happiness is his optimism and hope about the future. If he thought that the bottom could fall out at any time, he might not be so darn happy in the present. But he believes that the Lord will never desert him, and vice versa.