Learn Hardcore Russian: Подойди к столу и возьми карандаш.

подойди is the perfective imperative of подойти (“to approach”). It tells someone to perform the action once and bring it to completion: “go up/come over.”
подходи is the imperfective imperative of подходить, suggesting a repeated or ongoing action (“keep coming over” or “approach habitually”). In this instruction you want the learner to approach the table one time, so the perfective подойди is appropriate.

Why is к столу in the dative case?

The preposition к (“towards” or “to”) always governs the dative case in Russian. It indicates movement in the direction of something. That’s why стол (nominative) becomes столу (dative).

What exactly does the preposition к convey in подойди к столу?

Here к means “to” or “towards” in the sense of moving closer. So подойди к столу literally means “move up to the table” or “come over to the table.”

Why is возьми used instead of бери?

возьми is the perfective imperative of взять (“to take”), implying a single completed action: “take (once).”
бери is the imperfective imperative of брать, which can imply repeated or habitual taking. For a one-off instruction (“take the pencil now”), возьми is more natural.

What case is карандаш in, and why isn’t it changed?

карандаш is in the accusative case as the direct object of возьми. Since карандаш is an inanimate masculine noun, its accusative form is identical to its nominative form (карандаш).

How would you make this instruction polite or address more than one person?

You switch to the plural/formal imperative endings: Подойдите к столу и возьмите карандаш. Both verbs change: подойдите and возьмите.

Why aren’t there any words for “a” or “the” before столу and карандаш?

Russian has no articles (no direct equivalents of “a” or “the”). Context tells you whether you mean “a table,” “the table,” “a pencil,” or “the pencil.”

Can you reorder the sentence as Возьми карандаш и подойди к столу, and would it sound natural?

Grammatically you can swap clauses, but it sounds odd here. You normally approach the table first, then take the pencil. Saying Возьми карандаш и подойди к столу implies you’ll take the pencil before reaching the table, which breaks real-world logic.

How is Подойди к столу и возьми карандаш pronounced, and where are the stresses?

A rough pronunciation is: pa-DOY-di k sta-LOO ee voz-MEE ka-ran-DÁSH. The stressed syllables are:
• по-до́й-ди
• сто́-лу
• во-зь-ми́
• ка-ран-да́ш