The speaker of this poem is twenty years old.
He reveals his age in the second stanza:
Now, of my threescore years and ten,
Twenty will not come again,
And take from seventy springs a score,
It only leaves me fifty more.
Recall that a "score" of anything is twenty of them. Remember Abraham Lincoln's mention of "four score and seven years ago" in his Gettysburg Address? He meant "eighty-seven years ago." However many scores you have, just multiply that by twenty, and add the rest: (4 x 20) + 7 = 87.
Getting back to the poem, what this second stanza means is, "Of the total seventy years that I'll probably get to be alive, I'll never again be twenty like I am now. So if you subtract twenty from seventy, I've only got fifty more years to enjoy life!" (Saying "threescore years and ten" is saying "seventy years:" [3 x 20] + 10 = 70.)
It's really important to read that whole stanza when you're trying to figure out how old the speaker of the poem is. If you stopped after reading the first two lines of the stanza, you might think that the speaker is seventy instead of twenty--you might misinterpret the first two lines as "Now since I'm seventy, I'll never again be twenty." But after you read all four of those lines, you understand that the first two lines are actually saying, "Now, since I probably only get seventy years of life, my current age of twenty is never going to happen again."
You might have guessed that, instead of someone who's barely out of his teens, it's more likely that an older gentleman of seventy would be out walking, looking at beautiful trees and thinking about how life is so short and valuable. But the speaker of this poem is actually a twenty-year-old doing just that!