First and foremost, paper ballots everywhere. No touchscreens.
Second, from printing time onward, all handling of ballots, whether marked or unmarked, must be done in the presence of representatives of at least two political parties. At no point are ballots left alone with individuals or only one party (this is something a lot of states could learn from us!). This eliminates many avenues for cheating by adding or removing ballots from the count.
The ballots themselves are Scantron, fill in the dots with ink. The state provides standardized pens for this.
All registered voters in a precinct are tracked in a paper roll. In order to get a ballot, voters must either be in the registration book, or apply for a provisional ballot (which requires identification). Names are marked off the registration roll as they get their ballots, and a count is kept. At the end of voting, the total count of the voting machines must match the total count of voters exactly. Any difference triggers a manual count.
Scantron machines are randomly spot-checked - some percentage of them will be hand-counted, and those counts checked against the machine results. Multiple mismatches across the district/state would trigger an election-wide hand count. This insures against cheating by altering the behavior of the machines. (This is also where touch-screen machines fall down, imho, and should be flatly banned! There is no manual guarantee of the integrity of the results in a touch-screen machine.)
Initial counts on election day are provisional, and need to be certified via additional checks before being finalized (including the possibility of recounts).
There's more, but these are the big ones, and you get the idea. This entire system depends on protecting the integrity of physical ballots, which as you can see, is pretty straightforward.