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Hello Kate and welcome to the forum! Switzerland offers many beautiful hiking regions. Some examples are:
- the Jungfrau region in the Bernese Oberland. Interlaken and Wilderswil are situated very centrally. Popular tourist resorts such as Wengen, Lauterbrunnen and Grindelwald are right at the foot of some famous hiking trails;
- the Engadine in Graubünden. St. Moritz is pretty but expensive. Instead, you could consider smaller towns such as Samedan, Celerina, Bever and La Punt;
- the Matter valley in Wallis. Its most famous resort is Zermatt, but again this is rather popular and expensive. It won’t take you a lot of time to get there from nearby towns such as Täsch and St. Niklaus.
You could take a look at the information on the areas mentioned above and see what appeals to you most. For each region, day trips and hiking tips are listed. You may want to take a look at the general hiking and safety tips as well.
As for the weather: this can vary in October from sunny and warm to cold weather with rain and even snow. At higher altitudes, the chance of snow will increase as autumn progresses. It’s best to check weather forecasts daily, for example through the reliable website of SF Meteo. Please note that some cable cars and cog trains may shut down for maintenance in the course of October. During the first half of the month, most mountain transportation will still be running though.
Have fun preparing your trip!