Learn Hardcore Chinese: wǒ xūyào huí qù, ránhòu dǎ diànhuà gěi wǒ de māma.

Both are correct. In Mainland usage, 给我妈妈打电话 is very common; in Taiwan, 打电话给我妈妈 is extremely common. In everyday speech, both are widely understood, so use either.

Is 我的妈妈 natural, or should it be 我妈妈?

For close family, native speakers usually drop 的: 我妈妈, 我妈, 我爸爸. Keeping 的 (我的妈妈) can sound more formal, emphatic, or used when the noun phrase is long or contrasted. In this sentence, 我妈妈/我妈 is more natural.

Why is 去 added after 回? Doesn’t 回 already mean “to return”?

回 is the verb “to return,” and 去/来 adds direction relative to the speaker:

回去 = go back (away from where the speaker is)
回来 = come back (to where the speaker is) So 回去 is correct if the “back” destination is not where the speaker currently is.

Do I need to say where I’m going back to (e.g., home) after 回去?

Not necessarily. 回去 can stand alone if context is clear. If you want to be explicit, add a place:

回家 (go back home)
回学校
回公司 You can also combine with direction: 回家去 (common in speech), but plain 回家 is most typical.

Is 然后 necessary? Could I use 再 instead, or say the two actions without any connector?

然后 = “and then” (narrates sequence; neutral).
再 = “then/next” (often plan-like or instruction-like; commonly used with 先).
Serial-verb without a connector is also natural: 回去打电话 = “go back and call.”

All three are fine; pick based on tone:

Neutral narration: 回去，然后…
Step-by-step plan: 先回去，再打电话
Concise: 回去打电话

Where do I put 了 if the actions already happened? What about 过?

For completed events, put 了 after each verb phrase:

我回去了，然后给我妈打了电话。 If narrating one completed sequence, many also say:
我回去以后给我妈打了电话。 Use 过 for past experience (ever done), not a specific one-time event:
我给我妈打过电话。 (I have called my mom before.)

Can I add 一下 to soften “call”? Where does it go?

Yes. You can say:

给我妈打一下电话 (perfectly fine)
More colloquial: 给我妈打个电话 (using a measure word sounds even more natural) Avoid placing 一下 after the whole phrase; it belongs right after the verb: 打一下.

Do I need a measure word with 电话? What’s the difference between 打电话 and 打个电话/打一通电话?

打电话 = make a phone call (general).
打个电话 = make a quick/single call (colloquial, very common).
打一通电话 = one call; 通 is a measure word often used for phone calls (quite common; sounds a bit more formal/standard than 个 in some regions).

Why is 打 used with 电话? Are there other verbs?

打电话 is the fixed collocation: 打 covers “dial/make” in this context. Alternatives:

拨电话/拨打电话 (dial/make a call) — more formal or used in instructions/announcements. Colloquial 打给他 is shorthand for 打电话给他 (common, especially in Taiwan). Avoid forms like 打给他电话.

Can I use a 把 sentence like 把电话打给我妈妈?

No. 打电话 is a verb-object set phrase and doesn’t work well with 把 in that way. Say:

给我妈妈打电话
打电话给我妈妈 Using 把电话打给… is unnatural.

Any pronunciation or tone-sandhi tips in this sentence?

给我: both are 3rd tone in citation, so sandhi turns the first into 2nd: pronounced like géiwǒ.
妈妈: first syllable mā (1st tone), second syllable is neutral: māma.
的 is neutral tone here: de.
电话 is 4th + 4th; keep the second falling clear: diànhuà.

Do native texts put spaces between words like this sentence does?

No. Standard Chinese writing does not use spaces between words. The segmentation shown is for learners. A natural written line would be:

我需要回去，然后打电话给我妈妈。

Could I replace 需要 with 要/得/必须? What’s the nuance?

需要: need (objective necessity; a bit formal/neutral).
要: want/will/need (context decides; here it can mean “need to” or “going to”).
得 (děi): have to/need to (stronger necessity).
必须: must (very strong, non-negotiable). All fit, with increasing force: 要 < 需要/得 < 必须. For casual obligation, 要/得 are very common.