Updating latest IOS, IOS 15.1

Yes, update to. the latest (iOS 15.1) is highly recommended, it should go seamlessly. Nothing will be lost but it is always a Good Idea to make a backup just in case. As far as that goes, consider what you already share into iCloud (photos, calendar, messages, mail are likely to be there for most users) and you can probably get away with the standard iCloud backup which only takes a few minutes. Then make sure you have WiFi for the download, and ideally charger power, just launch the update from Settings/General/Software Update.

You should be aware that upgrading from previous major versions (before 15.x) is a big download, and it may take longer than you expect to complete. Don't think it has failed just because you don't see anything much happening and certainly DO NOT be tempted to try and restart your iPhone unless it tells you to do that. Particularly with older devices the processor is not the fastest animal on the block and it needs time to decompress and validate each component as it installs. So... pick a time when you can leave the phone to update itself, maybe for an hour, maybe a little longer.