Swiss language confusion—Swiss German or German?

Which areas speak which is clear, there is a hard language boundary. One side everything in German, the other side everything in French (or Italian), with very few bilingual areas.
See map here: https://upload.wikimedia.org/wikipedia/commons/9/9f/Map_Languages_CH.png

German speaking: Zürich, Luzern, Bern, Berner Oberland (Lauterbrunnen, Mürren, Thun)
French: Geneva, Lausanne
Italian: The south side of the Alps, Lugano, Bellinzona.

Note the spellings: Luzern. Lucerne is the French spelling, which is also used in English (for historical reasons).
Signs and trains are always in the language of the place, you will only see road signs or trains going to Luzern, even if getting on the train in (French speaking) Geneva.
The inverse works, getting a train from Zürich to Geneva, it will list the French spelling (Genève), but the announcements will be in German (Genf), French (Genève) and if you are lucky English (Geneva).

German / Swiss German

The official language is official German, mostly the same as in Germany, with a few differences (analogous to the differences between US English and British English). All signs and printed matter (Menus, price lists) will be in official German.
Example of differences: In the US you have an elevator, in the UK it is a lift
In Germany you have a Fahrstuhl , in Switzerland you have a Lift.
In Germany you buy a Fahrschein to travel on a train, In Switzerland a Billette.

Swiss German is a spoken dialect. It varies greatly from place to place. German TV has to subtitle it, it is that difficult to understand.
If they hear you are a foreigner, they will switch to "proper German", or even switch to English.

Just remember "Grüetzi", or "Bonjour".

They all appreciate any effort. The Swiss French section starts in Geneva and almost makes it to Bern but not quite. Check the Cantonal (state) websites to see which language comes up first. Bern/Lucerne and farther east and south are Swiss German, but my great aunt never spoke anything but French there and did fine. South of the crest of the Alps it is Italian, or in the southeast Romanish around the Swiss National Park. Being polite is more important than being exact... their English will be better than your French//German/Italian. The most useful things are to learn to count Ein, Zwei, Drei, etc. Also use google maps.ch or .it to learn proper place names in the local language. For example in South Tirol (part of Italy since WW I) each town has two names, I.e. Bolzano/Bozen.

Of course Switzerland/Schweiz/Suisse uses CHF (Confederation Helvetica Francs) and Italy uses Euros, and you think in dollars.... get used to being shocked at prices for everything but wine and move on.

You will have a great time.

Either way, Gruetzi or Guten Tag, will work, and be appreciated. Gruetzi is never wrong, and displays more respect from a tourist for recognizing the difference (IMO). Danke or merci will both work anywhere - they're smart people. America has dialects too, we just don't know it.

Chris F has explained it very clearly, as usual.

In Switzerland I always use Grüetzi, never Guten Tag, for the reasons Stan proposes. If there are two or more of you walking into a shop, restaurant, etc., you will hear them add an additional word; it is "Grüetzi mitenand" which translates roughly as "greetings among us" so includes more than one person. It is a nice thing to use when appropriate, and once you have heard it it will be easy to say it yourself. The accent is on the "mit" and the ending "d" is barely heard.

For "thank you" I like merci vilmal, which sort of combines French and German and is used everywhere. Vielen Dank is fine too.

In mountain Inns and restaurants I have seen some menu items in Swiss German, especially the local speciality. You may have to ask what it is, as standard German may not give you a clue.

And fortunately the Swiss franc is almost exactly on par with the US dollar ( 100 CHF = $100.47) so no conversion math is necessary.

Prices are not bad if you buy food in a grocery store, but restaurant meals are quite pricy.

Thanks everyone for the very helpful and detailed responses. I truly do appreciate all the time taken to respond to the questions on this forum. I check it daily and have learned so much. You all are a great resource for trip planning. It will be my first trip out of the U.S. I just got my passport in the mail the other day. I am so excited as i have fallen in love with Switzerland and i haven't even been there yet!! Thanks again!!

Carrie, this is a link to the menu of a restaurant which does list in Schweizerdeutsch: https://www.gifthuettli.ch/file.php?file=86&lang=de
It has local specialities, and lists in 4 languages: Baseldeutsch (Basel German), Hochdeutsch (official German), French and English.
You will note the pre-amble is still in Hochdeutsch, they want the non-Swiss to understand as well. It has some local dishes, which don't have a name in English, the translation is not always helpful, you either know what it is, or try it anyway and get a surprise.

Thanks for the additional information Chris. I have been looking at some menus to get an idea of prices and more importantly the types of food offered. I'm a picky eater so i want to make sure i can find something to eat. I'd like to try a rosti, but i'm not sure i'll be more adventuresome than that. I have noticed that most of the menus in Switzerland provide descriptions in English. So, i should be able to identify what the foods are. I feel sheltered in my American bubble so i'm looking forward to the new experience. And i'm trying to learn as much as i can before i go so i'm not walking around like a lost tourist. I'm sure i'll have many more questions before April. Thanks again for the information!