One day is enough if you plan to get of the shuttle bus at most of the stops and do short hikes. If you want to do longer hikes than you probably need more time.
I would definitely suggest you to spend more than one day if you want to visit all the points and do hiking. I have just visited zion last week. Unfortunately the angels landing and upper emerald pool trail were closed due to rock fall. According to the website its closed for indefinite period. I hope they will open the trails before your planned visit. We did observation point, hidden canyon and narrows in one day. If you start early in the morning then you can cover more and its comfortable to hike as well. Actually two days is enough to cover all the points and do hiking in the popular trails. About park permit, yes you need different permits for different parks. Each national park permit costs around $30. So if you are planning to visit more than two parks i will suggest you to buy annual permit for all the parks for $80. You can use it for any national park throughout the year.
Like others have mentioned, you can complete a few short hikes or one longer hike in a day. I recommend taking more than one day if you want to complete some of the longer hikes.
You'll definitely need more than one day if you want to fully enjoy each hike without becoming exhausted. We did The Narrows, but i've heard of people doing both The Narrows and Angel's landing in one day. I would at least give yourself a couple days.
Personally not sure. I have a senior pass that admits the vehicle. I only moved here in May too. But, I am told that Oct is the best weather. Guess heat let up, monsoon season is over & it's not cold.
If you are wanting to see all view points, it would take more than two days. You could probably do 3 hikes depending which ones in one day, you can buy a pass there for all national parks. Try and get there early to beat the crowd and the heat
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