How Swiss Voting Currently Works
Every resident must register with a municipal office when they move into or out of a municipality. The person's nationality is part of the registration. If you are not Swiss, you must of course have obtained a residence / work permit in the first place. This registration is checked by your landlord and employer, so you won't be able to legally rent a flat or work, or take advantage of any government services, if you don't register.
Voting typically happens several times (I think a max of 4 times) per year. It always takes place on a Sunday. Votes have to be in by 12 midday on Sunday. The results are often known within a few hours. There are only 8 million residents, of which about 1/3 are non-Swiss, so not a huge number of votes to count.
About a month before the day of a vote, ballot packets are mailed by the municipality to all the voters, i.e. all the Swiss, or depending on the canton, to even non-Swiss who are allowed to vote in some local elections. The packets contain paper ballot slips for federal, cantonal, and municipal elections and/or referendums. Elections happen every couple of years, like everywhere else. Referendums happen all.the.time.
The ballot card, on which your mailing address was printed, must be signed. The ballot slips can be filled out, placed into an inner envelope with no identifying information, and mailed back in the same pre-paid outer envelope -- you just turn the ballot card upside down so that the return address is shown in the envelope window.
Or, on voting day or the Saturday before, you can go to any of several local polling stations and personally hand over your ballot. The polling places and opening times are all conveniently listed on the ballot card.
I believe that the e-voting being tested is a "keeping up with the times" thing, with the view to someday supplant the paper ballots with voting over the Internet. I don't see why they'd want electronic voting machines in the polling places, because if they got rid of the paper ballots and didn't have voting over the Internet, then people would be forced to go to the polling stations, a big step backwards in convenience.
Further, I doubt that saving money, or getting faster results is the impetus. They have a proven system which already has fast results. If anything, it is being considered as a further convenience for the voters, who would no longer have to mark the ballots and carry the envelope to a post box or to the post office, and also wouldn't be confronted with having to show up at a polling place on Sunday if they procrastinated until it was too late to mail in the ballot. They could vote over the Internet up to the last minute.