Competent polling place workers are always in demand. If you are interested in becoming more involved in civic life and prefer such involvement to be concentrated into a couple of days rather than spread out over a much longer period, then working at a polling place on election day may be the right match for you.
Qualifications differ from state to state and even from locality to locality. But usually you have to be a registered voter in your county, be fluent in English, and not be a candidate for public office yourself. Prior to election day you are required to attend a class or seminar lasting several hours to learn about election procedures, polling place machinery, and laws regarding voting.
Duties are typically along these lines: Poll workers arrive at the polling place an hour before polls open to set things up. During voting hours, poll workers check voters’ names against a master list for that precinct, attempt to resolve problems regarding qualifications, issue ballots or allow access to voting machines, report malfunctioning devices to election authorities, prevent electioneering at the facility, make sure votes are cast in secret, and assist disabled voters with special needs when assistance is requested. After the last voter who arrived during legal voting hours has voted, the poll workers close down the facility and tally the votes using specified procedures. In localities where votes are digitally counted, the workers transmit results electronically to a central location. Paperwork is completed and election materials and equipment are secured and supervision is returned to the people who normally operate that particular facility. In many cases several of the poll workers are required to convey sealed envelopes holding official results, provisional ballots, and memory chips to a central receiving station in their part of the county.
Poll workers usually receive a stipend for their services. Depending on the county, it can range from car fare to over $200.
I have served in this capacity several times in Illinois. The hours are long (16 hours, 25 minutes was my longest shift) and activity at the polling place can alternate quickly between sluggish and hectic. But for one day you are a sworn official whose duty it is to ensure an honest and accurate election in that precinct. You may feel exhausted at the end of your shift but you get a great amount of satisfaction watching or listening to election results which you helped count. You and your coworkers made democracy possible. Every American should do this at least once.
A number of states have made it legally possible for select high school students to serve as poll workers even though they are not yet old enough to vote. Illinois is one of those states. In Illinois, a poll worker has the title election judge. The High School Student Judges program in suburban Cook County is representative of such programs in the state.
If you are interested in serving as a poll worker for the next election, now is a good time to look into it. If you can’t find information on your county’s web site, give the county government offices a call. Elections are usually administered by the county clerk or an election commissioner.
TL;DR Civics 
Another good place to start a search for poll worker requirements is the The U.S. Election Assistance Commission. They have compiled a PDF of Poll Worker Requirements by State.
Like you mentioned for Illinois, quite a few states now have opportunities to allow 16-17 year olds to get involved and most are listed in that document.
An extensive report on the problems involved in training and keeping competent poll works year over year is available at Millions to the Polls (PDF).
The introduction begins with:
The report finds that when students get involved in election work early, they say they are more likely to get involved again as adults.
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