For weather I use the Weather Channel app (paid to have ads removed). For my location it seems to be the most accurate and I like the detailed information it offers above and beyond the stock weather app.
For calendar I used Week Cal for several years, but recently switched to BusyCal for iOS since I also use BusyCal on my Mac. Honestly, Week Cal is probably better in terms of use and layout, but either is an improvement over the stock calendar app.
For email I use Airmail on both iOS and macOS. I switched to it to get my Gmail pushed and it works fine.
For maps, I use the stock app but I've used Google Maps before and liked that as well.
Safari is still my favorite browser on iOS (which is saying something because it's my least favorite browser on macOS). Firefox, Chrome and Dolphin are okay. I loved Atomic Browser but it's no longer developed.