Voting in general elections requires that you are a US citizen. Rumors will, that in some local elections you can vote if you pay taxes but I have still to figure out where those localities are.
To register as a voter you go to the local office of the Department of Motor Vehicles. You can register yourself as belonging to one of the parties or as an independent.
Typically election day is twice per year, spring and fall (although when there are primaries for presidential elections these may be separate events.) The international focus is naturally on the presidential election that takes place first Tuesday in November every 4 years. At this and the midterm elections a third of the senate, on a 6 year schedule, and all members of the House of Representatives are up for reelection. There are no term limits.
But each of the states also have to chose their governors and vice governors, state senators, state assembly members, state treasurer, mayors and vice mayors, members of the city councils, sheriffs, judges, district attorneys, head of police departments, district coroners, school council members, water district boards, banking and insurance commissioners... at regular intervals. Americans are highly surprised when learning that many of these positions are filled by appointed, not elected, officials in other countries.
Significant continous effort is expended for drawing attention to officials or propositions up for a vote. Much of this effort is concentrated around raising funds so the politician can buy TV commercial space, print signs, or send out flyers. Some fundraising takes place by asking people directly for money, some by having events where donors pay an exorbitant price for access to the politician. Enjoy dinner with Mr./Ms.Candidate at the Hampton's: $50.000. For another 10K you can be photographed with him/her and get a signed picture. Obviously included in the price is a place on the future White House short list.
For a while around 2004 we regularly received pictures of president Bush in the mail with invitations to support the Grand Old Party financially. It is a little difficult to discern how we ended up on that mailing list. Seen from the U.S., Scandinavia - even the conservative part - is way out on the socialist spectrum because not even conservatives question the wisdom of public healthcare and general access to higher education. The confusion may be exacerbated by linguistics: In Europe we primarily use the term liberal in context with support for free markets (where Americans would use libertarian. In Europe we hardly have anything as right winged as U.S. libertarians.) Americans use liberal about attitudes to social questions (abortion, gay marriage...) and liberals are generally found on the left wing of the Democrats.
Along the same line is the choice of colors. States where Republicans (the GOP, conservative, symbol: Elephant) are in majority are talked about as the Red States. (Democratic states are Blue States, symbol: Donkey). So don't assume that red and communist are synonyms. "Rather dead than Red" - the slogan from the early years of the Cold War - has a totally different connotation today. Should you wear a tie, assume that your choice of color may be analyzed. And not from a fashion viewpoint.
With all these people and perhaps propositions for pieces of legislation op for vote, you have to be prepared before you go to the voting booth. The time it takes to fill out the bubbles on the ballot is already a strain for the system and consequently a bigger and bigger part of the electorate vote by write in ballots.
All elected officials have to accept that not only is it open season all year for the press to hunt them; their employees may also decide to run against them and any mistake they make may be used against them at the next election. You should not only be competent at your job, you also have to be very good at marketing - the product being yourself and your agenda. If this latter requirement gets us the best and most competent officials is another question. I would think that being a good coroner could require a personality with love for details more than that of a public entertainer.