Eu vejo um castanheiro no jardim.

In Portuguese the verb ending already indicates the subject (here -o in vejo means “I”). So eu (“I”) is not strictly necessary and is often omitted in everyday speech:
• Vejo um castanheiro no jardim.
You include eu only for emphasis or clarity.

Vejo is the present indicative, first‐person singular form of the verb ver (“to see”).
• ver (infinitive) → eu vejo (I see)
It tells you who is doing the action (I) and when (right now).

Um castanheiro uses the indefinite article um (“a”) because you’re introducing one chestnut tree, not a specific one already known to the listener.
• If you had already mentioned or both know the tree, you’d say o castanheiro (“the chestnut tree”).
• You generally cannot drop the article before singular countable nouns, so vejo castanheiro sounds unnatural.

Castanheiro is a masculine singular noun. To make it plural, change -o to -os:
• Singular: um castanheiro
• Plural: dois castanheiros (“two chestnut trees”)
And if you use the definite article it becomes o castanheiro → os castanheiros.

No is the contraction of em + o (“in + the” masculine).
• em o jardim → contracted to no jardim
You’d use na (em + a) only if the noun were feminine, e.g. na sala (“in the living room”).

• ver = “to see,” passive perception (your eyes register something).
• olhar = “to look (at),” an intentional action: you direct your gaze.
Example:
• Eu vejo um castanheiro. (I see a chestnut tree.)
• Eu olho para o castanheiro. (I look at the chestnut tree.)

Yes. Drop the article if you mean chestnut trees in general (plural, unspecified):
• Eu vejo castanheiros no jardim. = “I see (some) chestnut trees in the garden.”
If you keep the article: Eu vejo os castanheiros no jardim. = “I see the chestnut trees in the garden” (those we both know).

Yes. Portuguese allows some flexibility for emphasis or style.
• No jardim vejo um castanheiro.
focuses first on where you see it.
Word order changes are fine as long as you keep subject–verb agreement and clarity.