Learn Hardcore Korean: choegeune naneun undongeul dasi sijakhaesseo.

최근에 means "recently" (a point or short window in the recent past). It pairs naturally with past-tense events like -었/았-.
요즘/요새 means "these days" (an ongoing period around now) and often goes with present/habitual statements.
Your sentence with 최근에 highlights the past starting point. 요즘 운동을 다시 시작했어 is also heard if you frame the restart as part of what’s going on these days, but 최근에 is the clearest for a recent, completed start.

Why is it 나는 and not 저는?

나/나는 is informal/casual; 저/저는 is polite/humble.
The ending -했어 is casual, so 나는 … 했어 is internally consistent.
In polite speech, use 저는 … 했어요/했습니다. Avoid mixing styles like 나는 … 했어요 in most situations.

Can I drop 나는?

Yes. Korean often omits the subject when it’s clear from context. 최근에 운동을 다시 시작했어 is perfectly natural in conversation.

What does 을 do after 운동, and when would I use 를?

을/를 marks the direct object. Use 을 after a consonant (운동), 를 after a vowel (예: 책을 vs 차를).
With 시작하다, you need an object: 운동을 시작하다.
With 운동하다 (a compound verb), both 운동하다 and 운동을 하다 are fine.
In casual speech, the object marker can be dropped if unambiguous: 운동 다시 시작했어.

Where should 다시 go? Is 다시 운동을 시작했어 different from 운동을 다시 시작했어?

Both are correct and mean the same thing. Natural orders:

최근에 다시 운동을 시작했어 (very common)
최근에 운동을 다시 시작했어 (also common) Placing 다시 close to the verb phrase (… 시작했어) is the norm.

Does 다시 시작했어 imply I had stopped before?

Yes. 다시 signals a return to a previous state: you did it before, stopped, and resumed. Be careful with 또; while it can mean "again," it often feels like "yet again/another time" and can sound irritated or repetitive. For resuming, 다시 is the default.

Does 시작했어 mean I’m still exercising now?

Not necessarily. It’s a simple past. Context often implies you’re continuing, but to be explicit:

Continuing: … 시작했고 지금도 하고 있어. / … 시작해서 계속 하고 있어.
Past/finished or remote: … 시작했었어 (had started; often implies it’s no longer the case or is backgrounded in time).

Why 나는 and not 내가?

나는 uses the topic marker, framing "as for me…" (often contrastive or setting background).
내가 uses the subject marker, focusing on "I" as the doer, useful for answering "who?" or for emphasis.
Example nuance: In a general update, 나는 …; answering "Who started again?" → 내가 시작했어.

Do I need 에 in 최근에? Can I say 최근 운동을…?

최근에 (with 에) is the adverb "recently."
최근 without 에 is an adjective meaning "recent" before a noun, e.g., 최근 뉴스 (recent news).
최근 운동을 다시 시작했어 sounds like "the recent exercise…," which isn’t what you want. Use 최근에.

Can I use the pattern V-기 시작하다 here?

Yes. 최근에 다시 운동하기 시작했어 is natural and means the same thing. Two common options:

운동을 시작하다 (start exercise; noun object)
운동하기 시작하다 (start doing exercise; verb nominalization) Both are fine. You may also hear 운동하는 걸(것을) 다시 시작했어 in everyday speech.

Is 운동을 재개했어 okay?

Grammatically yes, but 재개하다 is formal/news-y ("to resume" in official contexts). In daily conversation, 다시 운동을 시작했어 or 운동을 다시 하게 됐어 sound much more natural.

Can I drop particles like 을 or 에 in casual speech?

을/를: Often dropped in speech if the meaning is clear: 최근에 운동 다시 시작했어.
에 in 최근에: Don’t drop it; without 에, the meaning changes (it becomes an adjective). Keep particles in writing or formal situations.

How is this pronounced naturally?

시작했어: The ㅎ often causes the preceding ㄱ to be aspirated, so it sounds like "si-jak-ae-sseo" with a [kʰ] feel; many learners hear it as something like "si-jak-kae-sseo." Also, 했어 sounds like "hae-sseo" (the ㅆ is a tense s).
운동을 links smoothly: "un-dong-eul" (no extra pause).
나는: "na-neun." Exact IPA isn’t necessary; just expect some assimilation and a crisp tense s in -쎠/써.

Can 나는 be contracted to 난?

Yes. 난 is a very common contraction in casual speech: 최근에 난 운동을 다시 시작했어.

What’s the effect of 운동은 다시 시작했어 (topic on the object)?

Using 은/는 on the object makes it contrastive: "As for exercise, I started again (but as for something else, maybe not)." For example: 운동은 다시 시작했어, 식단은 아직이야. If you just want a neutral statement, stick with 운동을.