Learn Hardcore Chinese: wǒ xūyào zhǎo wǒ de tóngshì, ránhòu yìqǐ qù chāoshì. - I need to look for my coworker, then go to the supermarket together.

On its own, 找 means “to look for” or “to go see (someone),” without implying success. If you want to emphasize the result (that you find them), use 找到/找见.

Process: 我需要找我的同事，然后一起去超市。 (I need to look for/go see my colleague…)
Result: 我需要找到我的同事，然后我们一起去超市。
You can also say: 等我找到我的同事，我们一起去超市。

Can I say 我要 instead of 我需要? What’s the difference?

需要 = “need,” focuses on necessity. Neutral/formal.
要 is broader: “want to,” “going to,” or sometimes “need to,” depending on context. It can sound more volitional or like a plan.
For strong obligation, prefer 得/必须. Examples:
我需要找我的同事 (I need to find my colleague – necessity)
我要找我的同事 (I want to / I’m going to find my colleague)

Do I have to use 的 in 我的同事? Can I say 我同事?

Both are acceptable.

我的同事 is neutral and a bit more explicit/formal.
我同事 is common in speech and a bit more casual. With close relationships (e.g., family terms), 的 is often dropped. With 同事, both forms are widely used.

Who is the subject of 一起去? Should I add 我们?

The subject is understood from context: it means “then we go together.” Chinese often omits the subject when it’s clear. For explicitness, you can say:

然后我们一起去超市。
Or specify with the partner: 然后跟我的同事一起去超市。

Where does 一起 go? Can I put it after the verb or at the end?

一起 is an adverb and goes before the verb:

Correct: 我们一起去超市。
Incorrect: ✗ 我们去一起超市 or ✗ 我们去超市一起

Should I use 跟/和 to say “with my colleague”?

You can make it explicit with either:

跟我的同事一起去超市
和我的同事一起去超市 Both are fine; 跟 is a bit more colloquial. Keep 一起 in there to convey “together.”

What’s the difference between 然后, 再, and 之后/以后 for “then/after”?

然后: “then/and then,” a general connector for sequence. Very common.
再: “then (do the next step),” often used with 先: 先…再…. You can also do 然后再… to stress the next step.
之后/以后: “after(wards),” more like a time expression; a bit more formal or structural. Examples:
先找同事，然后/再一起去超市。
找到同事以后/之后，我们一起去超市。

Would it be more natural to add 先 (“first”)?

Adding 先 often sounds very natural when describing steps:

我需要先找我的同事，然后再一起去超市。 Your original is also fine; 然后 already implies sequence.

Should I use 去 or 到 with 超市?

去超市 emphasizes going (movement toward the place). Very common.
到超市 emphasizes arrival at the place. Often used with a result or time: 到超市以后…, 到了超市…. Both are correct; 去超市 is the default in everyday speech.

Do I need 了 here to show future or completion?

No. Chinese doesn’t mark future with 了. Your sentence expresses a plan/need without aspect marking. Use 了 when stating completed actions or changes:

Completed: 我已经找到了我的同事，然后我们就一起去超市了。
Planning/sequence (no 了): 我需要找我的同事，然后一起去超市。

How do I add time and place naturally?

Typical order: Subject + time + (modal) + place + verb (+ object).

我明天要在公司找我的同事，然后我们一起去超市。 If you also specify the companion:
我明天要在公司找我的同事，然后跟他一起去超市。

How can I make 找 sound softer or more polite?

Add 一下 after the verb, or use 联系:

我需要找一下我的同事。
我需要联系一下我的同事。 This makes the request/action sound lighter.

Should I say 一个同事 if it’s not a specific colleague?

Yes. Use 一个 to indicate “a/one (colleague)”:

Specific: 我的同事
Non‑specific: 一个同事
One of my colleagues: 我的一个同事

Any pronunciation tips? Especially for 一起.

一起 is pronounced yìqǐ (the tone on 一 changes to fourth tone before a third‑tone syllable).
然后: rán hòu (2nd + 4th).
需要: xū yào (1st + 4th).
超市: chāo shì (1st + 4th).

Why are there spaces between the Chinese words here? Is that normal?

In standard Chinese writing, there are no spaces between words. The spaces in the provided sentence are pedagogical, to help learners see word boundaries and match pinyin.