Stanza 3 Summary

Get out the microscope, because we’re going through this poem line-by-line.

He took his vorpal sword in hand;
Long time the manxome foe he sought–

Now we see why there was a warning in the first place. Apparently, the son is going hunting for these creatures. Maybe they are disturbing the otherwise peaceful wood.
And what's a vorpal sword? Luckily we have the word sword to tell us that it's a weapon; vorpal is just a modifier here. But what a modifier! It's reminiscent of vortex, warp, and portal. This thing sounds almost like a light-saber. It's no ordinary sword, that's for sure.
So our hunter, the hero of the story, goes out into the wild for a long time, to seek (the past tense is sought) the horrible creatures his parent has described to him.
One last fiddly word: manxome. Sounds a little bit like a combination between noisome (i.e., "foul") and mangy. That would be about right, it would seem, for evil creatures that one wants to vanquish with a sword.
So in short: our hero takes his weapon, heads off, and spends a good long time trying to find these evil creatures.

So rested he by the Tumtum tree,
And stood awhile in thought.

Thankfully, these lines are starting to get a little bit easier to comprehend.
We might not know what a Tumtum tree is, but the rest of these lines are very clear.
Our hero has been seeking his foes for a while, and is now taking a break.
So he rests by the tree, and loses himself thinking for a little while.
He might be thinking about how best to find these scary beasts. Or how he might best conquer them. Or maybe just about how tired he is.