"I work on weekdays."
Hijacking the top comment to explain this for people in the future: はたらく = "to work". かいしゃ= "office". しごと = "job". In English we typically refer to all of these as "work" which is why it get get frustrating to parse them. This is why we we need a をし when we verb しごと. Because it literally saying "I am jobbing..." which is sounds moronic if done directly.
はたらくis the regular non-past form; this one actually means "I/you/he/she/we/you/they (work/will work)." はたらき is a stem which is used to make the -ます etc. forms and some other derived concepts, like the 働きたいです"I want to work" or the 働きそうです "he seems to be working" forms.
I think this Youtube video from "Japanese From Zero" provides an explanation for anyone confused about する: https://youtu.be/7ecX6VSSNxQ?t=244, I've timestamped the relevant point for this comment but the whole video seems good.
In the dictionary I'm looking at, it lists しごと (仕事) as a "suru verb"-- "する" can be applied directly to the noun "work" to convert it into the verb "to work", and "します" is the masu form of suru, so しごとします seems correct. That being said, 仕事をします isn't incorrect per se as it also means "do work". So I think both should be correct, AFAIK; which you'd actually use would come down to what's more commonly said in Japan.
My dictionary lists it as a noun, but does provide a list of conjugations. I've never heard it used as anything but a noun but given my level there's not a lot that i have heard. It is possible that the Duolingo writers want us to use it as a noun exclusively and follow the standard "noun-to-verb conversion" pattern of ['noun' を し-'masu-ending']. It is possible that this is actually an example of noun-to-verb conversion used solely in order to teach verb conversion.
"I work on weekdays." literally translates to "平日に働きます。" This answer is accepted. (April 22, 2021) Just want to add that I am grateful for the length and depth of the discussion in this thread. It makes me better appreciate the scope of the language even for a sentence as simple as this.
See my reply to TomosFrate further up the page. As far as I'm aware: in this case there should be no difference, but if the noun isn't a "suru verb" then you would need to say "をします" or "をする". Even then, there would be a preferred way of saying it in Japanese, but it's seemingly case-by-case for each word.
週末 = しゅうまつ = End of the week / weekend
平日 = へいじつ = Regular day / weekday (not weekend or holiday)
Those two are the most common AFAIK, though there are other words with similar meanings. E.g. 週日 = しゅうじつ = Weekdays, including any holidays (possibly including Saturday?) which seems to be less commonly used.
There is no subject specified in the sentence (so I is implied). 平日 is the topic. Any part of a sentence can become the topic. With a subject specified the sentence would be something like: 平日は私が仕事をします。 What might confuse you is that in two special bot also the most common cases, the particle before は particle is omitted. This is true for が (subject particle) and を (object particle). But は alone (without other particle before) doesn't mean that it is either subject or object. You can also make some part of the sentence the topic that doesn't need a particle. This is what happened in this case. The time (on weekdays) is the topic you're talking about, and the information about this topic is that you do work. So you were right that the subject is an implied "I", even in this sentence.