There are a lot of free weather apps, I would try several of them because some may not support your area. To pin it on the main screen, find the app name in the list. Press and hold the app name, and choose the "pin to start" option.
There are lots of good Nokia only apps in the Nokia collection (Nokia Maps has an offline option!)
Other apps I use (I'm Canadian, I don't know if they are available where you live):
1. Fantasia Painter Free - good for drawing or photo editing
2. Metrotube - the best Youtube app
3. Supertube - not as good as Metrotube but you can download videos and watch offline
4. Taptitude - lots of fun games, all free
5. Xbox LIVE Extras - customise your Xbox avatar
6. UC Browser - my favourite web browser, it has forward and backward buttons
7. Adobe Reader for pdf files. It's much better after the new update
8. Wikipedia (the one by Rudy Huyn) - beautiful and functional, also can save articles to read offline
Other random things:
1. If have 45 or more apps, you get letters in your app list. Tap on a letter to jump quickly to a another part of the list.
2. If you press and hold the area you are typing in, you get a cursor that you can move.
3. Single tap a word to highlight it. There are two bars you can drag to select more text. A button to copy text will also appear. The paste button appears on the keyboard.
4. You can pin people to the main screen too. Press and hold on a contact's name in the People hub. Then choose "Pin to start".